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Absolute Irreducibility of Polynomials via Newton Polytopes
Journal of Algebra 237, 501᎐520 Ž2001. doi:10.1006rjabr.2000.8586, available online at http:rrwww.idealibrary.com on
Absolute Irreducibility of Polynomials via Newton Polytopes Shuhong Gao Department of Mathematical Sciences, Clemson Uni¨ ersity, Clemson, South Carolina 29634 E-mail: [email protected] Communicated by Walter Feit Received July 12, 1999
A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping a certain collection of them nonzero. 䊚 2001 Academic Press Key Words: fields; multivariable polynomials; absolute irreducibility; Newton polytopes; Newton polygons; indecomposable polytopes; Eisenstein criterion; Minkowski sums.
1. INTRODUCTION It is well known that Eisenstein’s criterion gives a simple condition for a polynomial to be irreducible. Over the years this criterion has witnessed many variations and generalizations using Newton polygons, prime ideals, and valuations; see for example w3, 25, 28, 38x. We examine the Newton polygon method and generalize it through Newton polytopes associated with multivariable polynomials. This leads us to a more general geometric criterion for absolute irreducibility of multivariable polynomials. Since the Newton polygon of a polynomial is only a small fraction of its Newton polytope, our method is much more powerful. Absolute irreducibility of polynomials is crucial in many applications including but not limited to finite geometry w14x, combinatorics w47x, algebraic geometric codes w45x 501 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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permutation polynomials w23x, and function field sieve w1x. We present many infinite families of absolutely irreducible polynomials over an arbitrary field. These polynomials remain absolutely irreducible even if their coefficients are modified arbitrarily but with a certain collection of them nonzero. As in many standard algebra textbooks, Eisenstein’s criterion w5x is described as follows. EISENSTEIN’S CRITERION. Let R be a unique factorization domain and let f s f 0 q f 1 X q ⭈⭈⭈ qf n X n g Rw X x. If there is a prime p g R such that all the coefficients except f n of f are di¨ isible by p, but f 0 is not di¨ isible by p 2 , then f is irreducible o¨ er the fraction field of R. Several people ŽDumas w4x, Kurschak w22x, Ore w30᎐32x, Rella w35x. have generalized this criterion by using Newton polygons. Assume that f 0 f n / 0. One can construct a polygon in the Euclidean plane as follows. Suppose that the coefficient f i is divisible by p a i but not any higher power, where a i G 0 and a i is undefined if f i s 0. Plot the points Ž0, a0 ., Ž1, a1 ., . . . , Ž n, a n . in the Euclidean plane and form the lower convex hull of these points. This results in a sequence of line segments in the first quadrant of the plane, called the Newton polygon of f Žwith respect to the prime p .. Dumas w4x determines the degrees of all the possible nontrivial factors of f in terms of the widths of the line segments on the Newton polygon of f. Consequently a simple criterion for the irreducibility of f is established. EISENSTEIN ᎐DUMAS CRITERION. Let R be a unique factorization domain and let f s f 0 q f 1 X q ⭈⭈⭈ qf n X n g Rw X x with f 0 f n / 0. Assume that f is primiti¨ e; i.e., f 0 , . . . , f n ha¨ e no nontri¨ ial common factor in R. If the Newton polygon of f with respect to some prime p g R consists of the only line segment from Ž0, m. to Ž n, 0. and if gcdŽ n, m. s 1 then f is irreducible in Rw X x. The condition on the Newton polygon means that a i G Ž n y i . mrn for 0 F i F n where p a i exactly divides f i . When m s a0 s 1, this condition is the same as in Eisenstein’s criterion. Hence the Eisenstein ᎐Dumas criterion generalizes that of Eisenstein. The Eisenstein ᎐Dumas criterion was originally proved for integer coefficients. Later it was generalized to local fields or any field with valuations w3, 20, 25x. We are interested in the case when R is a polynomial ring over a field. Let F be a field and let R s F w Y x where Y is a new variable. Then Y is a prime in R and the Eisenstein ᎐Dumas criterion can be applied in Rw X x ( F w X, Y x. We restate the criterion as follows. EISENSTEIN ᎐DUMAS CRITERION Ža special case.. Let F be any field and let f s f 0 Ž Y . q f 1Ž Y . X q ⭈⭈⭈ qf nŽ Y . X n g F w X, Y x. Assume that f 0 Ž Y . / 0
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and f nŽ Y . is a nonzero constant in F. If the Newton polygon of f Ž with respect to Y . has only one line segment from Ž0, m. to Ž n, 0. and gcdŽ n, m. s 1, then f is Ž absolutely . irreducible o¨ er F. A polynomial over a field F is called absolutely irreducible if it remains irreducible over every algebraic extension of F. The same proof for the irreducibility of f under the Eisenstein ᎐Dumas condition also shows that f is absolutely irreducible. The above criterion was also discovered by Wan w48x. In w36, Theorem 1B, p. 92x, Schmidt describes another method for constructing absolutely irreducible polynomials which he attributes to Stepanov w43, 44x. This method can also be interpreted as a polygon method. Let f s f 0 Ž Y . q f 1Ž Y . X q ⭈⭈⭈ qf nŽ Y . X n g F w X, Y x. The upper Newton polygon of f with respect to Y is defined to be the upper convex hull of the points Ž0, a0 ., Ž1, a1 ., . . . , Ž n, a n . where a i is the degree of f i Ž Y . in Y and a i is not defined if f i Ž Y . s 0. STEPANOV ᎐SCHMIDT CRITERION. Let F be a field and let f g F w X, Y x with degree n in X. If the upper Newton polygon of f with respect to Y has only one line segment from Ž0, m. to Ž n, 0. and gcdŽ n, m. s 1, then f is absolutely irreducible o¨ er F. Note that the Eisenstein ᎐Dumas and Stepanov᎐Schmidt criteria read exactly the same except that they exploit ‘‘different parts’’ of the polynomials. This leads us to consider the convex hull of the exponent vectors Ž i, j . of all the nonzero terms cX i Y j of a polynomial f and call the resulted convex set the Newton polytope of f. Its boundary gives us a ‘‘whole’’ polygon that contains both the lower and upper polygons used above. This concept of Newton polytopes associated with polynomials is due to Ostrowski Ž1921. and is similarly defined for any multivariable polynomials. Ostrowski realizes that the factorization of polynomials implies the decomposition of polytopes in the sense of the Minkowski sum. In the 1970s, Ostrowski wrote two papers w33, 34x dealing with term ordering and irreducibility of multivariable polynomials. His irreducibility criteria are, however, based mainly on algebraic techniques Žsuch as algebraic independence. and Puiseux developments. We show that Newton polytopes carry a lot of information about the irreducibility of polynomials. Indeed, the Eisenstein ᎐Dumas and Stepanov᎐Schmidt criteria are just very special cases of our results. More precisely, we study the irreducibility of multivariable polynomials through the decomposability of their Newton polytopes. Our main contribution is in the construction of indecomposable polytopes and thus we give many classes of absolutely irreducible polynomials over an arbitrary field.
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To get a glimpse of our results, we give two examples here; more general results can be found in Section 4. EXAMPLE 1. Let F be any field and let f s aX n q bY m q cX u Y ¨ q
Ý ci j X i Y j g F w X , Y x
with a, b, c nonzero. Suppose that the Newton polytope of f is the triangle with vertices Ž n, 0., Ž0, m., and Ž u, ¨ .. If gcdŽ m, n, u, ¨ . s 1 then f is absolutely irreducible over F. EXAMPLE 2. Suppose that the Newton polytope of f s a1 X l q a2 Y m q a3 Z n q a4 X u Y ¨ Z w q
Ý c i jk X i Y jZ k g F w X , Y , Z x
is the tetrahedron with vertices Ž l, 0, 0., Ž0, m, 0., Ž0, 0, n., and Ž u, ¨ , w .. Then f is absolutely irreducible over F, provided gcdŽ l, m, n, u, ¨ , w . s 1. Our paper is organized as follows. In the next section, we define the decomposability of polytopes with respect to Minkowski sums and discuss its relation with the factorization of polynomials. A general irreducibility criterion is established. Note our concept of decomposability of polytopes is incompatible with that in the literature, say in Grunbaun’s book w13, ¨ x Chap. 15 . In Section 3, we collect properties of Minkowski sums of polytopes, particularly on the decomposition of faces of polytopes. In Section 4, we give two general constructions of indecomposable polytopes and thus give many simple and explicit criteria for absolute irreducibility of multivariable polynomials. Many infinite families of absolutely irreducible polynomials are described. Related Works We should mention that Lipkovski w24x associates a polynomial with a polyhedron Žunbounded., called its Newton polyhedron, which is a direct analogue of Newton polygon in higher dimension. Indecomposability of its Newton polyhedron implies the analytic irreducibility of a polynomial at the origin Ži.e., irreducibility in the formal power series ring.. To that extent, Lipkovski’s method is local while our polytope method is somewhat global. Lipkovski discusses indecomposability of polyhedra and gives several constructions of indecomposable polyhedra. Filaseta w7x uses the Newton polygon method to decide irreducibility of Bessel polynomials. Wan w49x uses the Newton polygon to study zeta functions and L functions. Gao and Shokrollahi w10x use the Newton polygon method to compute roots of polynomials over unction fields of curves. For a survey of the Newton polygon method, see Mott w28x.
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There are several other methods in the literature for proving absolute irreducibility. One is using singularity analysis: if a polynomial defines a smooth hypersurface then it is absolutely irreducible. This method is often used in algebraic geometry. Another method is presented by Janwa et al. w15x using Bezout’s theorem on intersection multiplicity of curves. Noether’s irreducibility forms w29x give yet another powerful method for proving irreducibility of polynomials. Noether’s forms are carefully analysed by Schmidt w36x and greatly improved by Kaltofen w19x. For efficient algorithms for testing irreducibility of multivariable polynomials, see Gao w9x, von zur Gathen w11x, and Kaltofen w17, 18x. We should also mention that Newton polytopes have been used extensively to study toric ideals and solutions of systems of Žmultivariable. polynomial equations; see Gel’fand et al. w12x, Khovanskiı˘ w21x, Sturmfels w46x, and the references therein.
2. POLYTOPE METHOD We assume that the reader is familiar with the basic properties of w13x, Webster w50x, and Ziegler w52x. polytopes; see Ewald w6x, Grunbaun ¨ For the convenience of the reader, we review some basic concepts that will be needed in the sequel. Let ⺢ be the set of real numbers and let n be a positive integer. A subset S : ⺢ n is called convex if, for any two points a, b g S, the line segment from a to b is also contained in S; that is, a q Ž b y a . s Ž 1 y . a q b g S,
᭙0 F F 1.
For any subset S : ⺢ n, convŽ S . denotes the smallest convex set in ⺢ n that contains S. It is straightforward to check that conv Ž S . s
½
t
t
Ý i ai : ai g S, i G 0,
Ý i s 1
is1
is1
5
.
When S s a1 , . . . , a k 4 is a finite set, denote convŽ S . by convŽ a1 , . . . , a k ., which is called the convex hull of a1 , . . . , a k . The convex hull of finitely many points is called a polytope. A point of a polytope is called a vertex if it is not on the line segment of any other two points of the polytope. It is well known that a polytope is always the convex hull of its vertices. We consider polynomials with n variables X 1 , X 2 , . . . , X n . Let F be any field and let f s Ý f i1 i 2 ⭈ ⭈ ⭈ i nX 1i1 X 2i 2 ⭈⭈⭈ X ni n g F w X 1 , X 2 , . . . , X n x. An exponent vector Ž i1 , i 2 , . . . , i n . of f can be considered as a point in ⺢ n. The
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Newton polytope of f, denoted by Pf , is defined to be the convex hull in ⺢ n of all the points Ž i1 , i 2 , . . . , i n . with f i1 i 2 ⭈ ⭈ ⭈ i n / 0. For two sets A and B in ⺢ n, define A q B s a q b : a g A, b g B4 , which is called the Minkowski sum of A and B. LEMMA 2.1 ŽOstrowski w33x.. gh. Then Pf s Pg q Ph .
Let f, g, h g F w X 1 , X 2 , . . . , X n x with f s
Proof. This result is well known in the literature. For the sake of completeness, we give a simple proof here. By multiplication of polynomials, it is obvious that Pf : Pg q Ph . To prove the other inclusion, let v be any vertex of Pg q Ph . We show that there are unique points v1 g Pg and v2 g Ph such that v s v1 q v2 . Since v g Pg q Ph , the existence is no problem. Suppose that there is another pair v1X g Pg and v2X g Ph such that v s v1 q v2 s v1X q v2X .
Ž 1.
Then vs
1 2
Ž v1 q v2X . q
1 2
Ž v1X q v2 . .
Since v1 q v2X , v1X q v2 g Pg q Ph and v is a vertex of Pg q Ph , one must have v1 q v2X s v1X q v2 .
Ž 2.
Subtracting Ž2. from Ž1. yields v2 y v2X s v2X y v2 ,
i.e., 2 Ž v2 y v2X . s 0.
Hence v2 s v2X and v1 s v1X . Since v is a vertex of Pg q Ph , v1 and v2 must be vertices of Pg and Ph , respectively. There is a unique term in the expansion of g ⭈ h that has v as its exponent vector. Hence v g Pf . This proves that all the vertices of Pg q Ph are in Pf . Consequently, Pf = Pg q Ph as a polytope is the convex hull of its vertices. A point in ⺢ n is called integral if its coordinates are integers. A polytope in ⺢ n is called integral if all of its vertices are integral. Certainly, Newton polytopes of polynomials are integral. An integral polytope C is called integrally decomposable if there exist integral polytopes A and B such that C s A q B where both A and B have at least two points. Otherwise, we say that C is integrally indecomposable. Note that our concept of indecomposability is different from that in Grunbaun’s book ¨ w13, Chap. 15x; see the comments at the end of the paper. Since we will not encounter any other type of decomposability in this paper, the word ‘‘integrally’’ will be freely omitted in the sequel.
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IRREDUCIBILITY CRITERION. Let F be any field and let f g F w X 1 , X 2 , . . . , X n x be a nonzero polynomial not di¨ isible by any X i . If the Newton polytope of f is integrally indecomposable then f is absolutely irreducible o¨ er F. Proof. First note that f has no factor with only one term. Suppose that f factors nontrivially over some algebraic extension of F, say f s gh, where both g and h have at least two nonzero terms. Then the Newton polytopes of g and h have at least two points. By Lemma 2.1, Pf s Pg q Ph , contradicting our assumption that Pf is integrally indecomposable. When Pf is decomposable, f can be either reducible or irreducible. For example, if f s 1 q Y q XY q X 2 q Y 2 then Pf is decomposable Žequal to the sum of the triangle Ž0, 0. ᎐ Ž1, 0. ᎐ Ž0, 1. with itself.. Over a field F of characteristic different from two, it can be verified directly that f is absolutely irreducible. Over a field F of characteristic two, however, we have f s Ž1 q X q Y . Ž1 q X q 2Y . , where is an element of order 3 Žso f is irreducible over F if f F .. It remains to show that indecomposable Newton polytopes exist; thus absolutely irreducible polynomials can be constructed via the irreducibility criterion above. This will be our main focus in Section 4. We conclude this section with some comments. Remarks. Ž1. One can change the coefficients of a polynomial f arbitrarily and its Newton polytope will remain the same provided the coefficients of all the terms of f that correspond to vertices are nonzero. If the Newton polytope of f is indecomposable then f will remain absolutely irreducible when its coefficients are modified arbitrarily but with those of vertices nonzero. This gives great freedom in choosing suitable polynomials in applications. For all the examples in the sequel, we often give polynomials with coefficients fixed at 1, but one may change the coefficients to any nonzero elements in the ground field. Ž2. In w33x, Ostrowski uses the term ‘‘baric polyhedron’’ in place of the ‘‘Newton polytope’’ of a polynomial. Lemma 2.1 is proved for more general polynomials called algebraic polynomials where the exponents of variables may be rational numbers. Ostrowski mentions that he gave a talk at a German Mathematical Society meeting in 1921 about the baric polyhedron and its applications to the irreducibility problem. In his sequel paper w34x, Ostrowski discusses irreducibility of polynomials in details. It is surprising, however, that the concept of decomposability of polytopes does not arise there. Ostrowski develops irreducibility criteria using mainly
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algebraic tools Žsuch as algebraic independence and conjugates. and the Puiseux developments. Ž3. For the case of polynomials with three variables, the concept of the Newton polytope also appears in Shanok’s paper w39x. Shanok develops irreducibility criteria for polynomials by projecting a Newton polytope from ⺢ 3 into planes. Ž4. As mentioned in the introduction, Lipkovski w24x develops an analogue of the Newton polygon method in higher dimension for formal power series. Lipkovski associates a power series f of n variables with a Newton polyhedron Pf q ⺢ 0n , where Pf is defined similarly to polynomials and ⺢ 0 is the set of nonnegative real numbers. A Newton polyhedron is unbounded and, when n s 2, its finite edges form the Newton polygon. Lipkovski defines a similar concept of decomposability of Newton polyhedra and gives several constructions of indecomposable Newton polyhedra. For a polynomial of two variables, the only indecomposable Newton polyhedron is that corresponding to the Newton polygon as described in Eisenstein ᎐Dumas criteria. We will see, however, that there are many indecomposable polytopes in dimension two or higher.
3. SOME PROPERTIES OF POLYTOPES To further discuss the decomposability of polytopes, we need more properties about Minkowski sums of polytopes, particularly on the decomposition of their faces. Minkowski sums of convex sets have been extensively studied in the literature; see for example Schneider w37x. Let P be a polytope in ⺢ n. A face of P is by definition the intersection of P with a supporting hyperplane to P. In other words, a face of P is the set of all the points in P that maximize some linear function. A vertex is a just face of dimension 0. A face of dimension 1 is a line segment, called an edge of P. A face of dimension 1 less than that of P is called facet of P. The next result describes how faces decompose in a Minkowski sum of w13, polytopes; for its proof, see Ewald w6, Theorem 1.5x, Grunbaun ¨ Theorem 1, p. 317x, or Schneider w37, Theorem 1.7.5x. LEMMA 3.1.
Let A and B be polytopes in ⺢ n and let C s A q B.
Ža. Each face of C is a Minkowski sum of unique faces of A and B. Žb. Let C1 be any face of C and let c1 , c 2 , . . . , c k all of its ¨ ertices. Suppose that c i s a i q bi where a i g A and bi g B for 1 F i F k. Let A1 s conv Ž a1 , a2 , . . . , a k . ,
B1 s conv Ž b1 , b 2 , . . . , bk . .
Then A1 and B1 are faces of A and B, respecti¨ ely, and C1 s A1 q B1.
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Note that part Žb. is the constructive version of part Ža. and it says that the decomposition of all the faces is determined by the decomposition of vertices alone. This is extremely useful in applications. Part Ža. can be strengthened as follows. Let C1 be any face of C. Suppose that A1 and B1 are any convex subsets of A and B, respectively, such that C1 s A1 q B1. Then A1 and B1 must be faces of A and B, respectively. The proof is a little subtle and will be given elsewhere. A polytope is associated naturally with a graph consisting of its vertices and edges. LEMMA 3.2 ŽBalinski w2x.. The graph of a polytope of dimension n is n-connected. A convex cone with a vertex v is defined to be a convex set S in ⺢ n such that v is an extreme point of S and, for any a g S, v q Ž a y v. g S for all real numbers G 0. The next result must be known in the literature, but we could not find a convenient reference, so a proof is included. LEMMA 3.3. Let C be a con¨ ex cone with ¨ ertex v and let H be a hyperplane in ⺢ n with v f H. Suppose that Q s C l H is nonempty and bounded. Then, for any r g ⺢ n, either C l Ž r q H . is empty or there exists a real number t G 0 such that C l Ž r q H . s v q t Ž Q y v. s v q t Ž a y v. : a g Q4 . Proof. Choose ␣ g ⺢ n and  g ⺢ such that H s x g ⺢n : ␣ ⭈ x s  4
and
␣ ⭈ v ) .
We show that for every point a g C with a / v,
␣ ⭈ a - ␣ ⭈ v.
Ž 3.
Suppose on the contrary that ␣ ⭈ a0 G ␣ ⭈ v for some a0 g C. Let b g Q s C l H be any fixed point. Then
␣ ⭈ b s  - ␣ ⭈ v F ␣ ⭈ a0 . Let a1 s 1 a0 q Ž1 y 1 . b where 1 s Ž ␣ ⭈ v y  .rŽ ␣ ⭈ a0 y  . ) 0. Since 1 F 1 and C is convex, we have a1 g C and
␣ ⭈ a1 s ␣ ⭈ v.
Ž 4.
For any t G 0, b q t Ž a1 y v . s v q Ž t q 1 .
žž
t tq1
a1 q
1 tq1
b yv
/ /
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SHUHONG GAO
belongs to C, as a1 , b g C and C is a convex cone with vertex v. By Ž4., ␣ ⭈ Ž b q t Ž a1 y v.. s ␣ ⭈ b s  . Hence b q t Ž a1 y v. g H and b q t Ž a1 y v. g C l H s Q for all t G 0, contradicting the boundedness of Q Žnote that a1 / v.. Therefore Ž3. holds. For any r g ⺢ n and any a g C with a / v, consider the intersection of the ray
v q Ž a y v . : G 04
Ž 5.
with the hyperplane r q H s r q x g ⺢ n : ␣ ⭈ x s  4 s x g ⺢ n : ␣ ⭈ x s ␣ ⭈ r q  4 . Ž 6. Note that ␣ ⭈ Žv q Ž a y v.. s ␣ ⭈ r q  implies that s Ž ␣ ⭈ v y  y ␣ ⭈ r .rŽ ␣ ⭈ v y ␣ ⭈ a.. Since ␣ ⭈ v y ␣ ⭈ a ) 0 by Ž3., G 0 iff ␣ ⭈ r F ␣ ⭈ v y  . When this condition is satisfied, r q H intersects every ray Ž5. at a unique point determined by above. For r s 0, since ␣ ⭈ v y  ) 0 s ␣ ⭈ r, each ray Ž5. intersects H, and thus Q, at a unique point. Hence we may index all the rays Ž5. by a g Q. Now suppose that ␣ ⭈ r F ␣ ⭈ v y  . Then for each a g Q, the ray Ž5. intersects r q H at the point b s v q 0 Ž a y v. where
0 s
␣⭈vyy␣⭈r ␣⭈vy
.
Therefore the lemma holds with t s 0 . 4. INDECOMPOSABLE POLYTOPES AND ABSOLUTELY IRREDUCIBLE POLYNOMIALS We now proceed to construct indecomposable polytopes in ⺢ n. Each type of indecomposable polytope gives us a family of absolutely irreducible polynomials. When a polytope has only one point we say that it is trivial. We examine several types of simple nontrivial polytopes such as line segments, triangles, tetrahedrons, pyramids, etc. We show how to construct indecomposable polytopes from a given polytope. We need more terminology. A line segment convŽv1 , v2 . is simply denoted by v1v2 . For an integral point or vector v s Ž a1 , . . . , a n ., we write gcdŽv. to mean gcdŽ a1 , . . . , a n ., i.e., the greatest common divisor of the components in v. Similarly, for several points v1 , . . . , vk , gcdŽv1 , . . . , vk . means that gcd of all the components in v1 , . . . , vk together. For example, if v1 s Ž n, 0., v 2 s Ž0, m ., and v 3 s Ž u, u ., then gcdŽv1 , v 2 , v 3 . s gcdŽ n, 0, 0, m, u, u. s gcdŽ n, m, u.. For any two integral vectors v1 and v2 , we have gcdŽv1 , v2 . s gcdŽv1 , v2 y t v1 . for any integer t.
IRREDUCIBILITY OF POLYNOMIALS
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LEMMA 4.1. Let v0 and v1 be two distinct integral points in ⺢ n. Then the number of integral points on the line segment v0 v1 , including v0 and v1 , is equal to gcdŽv0 y v1 . q 1. Further, if v2 is any integral point on v0 v1 , then
s
gcd Ž v2 y v0 . gcd Ž v1 y v0 .
,
where
0 F t F 1.
Since v0 is integral, v is integral iff t Žv1 y v0 . is integral. But the components of v1 y v0 are all integers, so t must be rational. Let ts
i k
,
for some 0 F i - k with gcd Ž k , i . s 1.
Then t Žv1 y v0 . is integral iff k < gcdŽv1 y v0 .. Hence if v is an integral point different from v0 and v1 , then t must be of the form ts
i d
,
0 - i - d,
where d s gcdŽv1 y v0 . G 1. The number of choices for i is d y 1. So the total number of integral points v on v0 v1 is d y 1 q 2 s d q 1. Suppose v2 s v0 q ird Žv1 y v0 . is any integral point on v0 v1 with 0 F i F d where d s gcdŽv1 y v0 .. Note that Žv1 y v0 .rd is integral and gcdŽŽv1 y v0 .rd . s 1. Hence gcdŽv2 y v0 . s gcdŽ i ⭈ Žv1 y v0 .rd . s i. Also
Therefore the equation in the lemma holds. THEOREM 4.2. Let Q be any integral polytope in ⺢ n contained in a hyperplane H and let v g ⺢ n be an integral point lying outside of H. Suppose that v1 , v2 , . . . , vk are all the ¨ ertices of Q. Then the polytope convŽv, Q . is integrally indecomposable iff gcd Ž v y v1 , v y v2 , . . . , v y vk . s 1. Proof. Let C s convŽv, Q . as depicted in Fig. 1. Suppose that C s A q B for some integral polytopes A and B in ⺢ n. By appropriately shifting A and B, we may assume that v g A and 0 g B. Note that v, v1 , . . . , vk are all
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FIG. 1. Indecomposable pyramid.
the vertices of C, and vv1 , . . . , vvk are edges of C. By Lemma 3.1, there are unique vertices a i g A and b i g B such that vi s a i q b i ,
1 F i F k,
and vvi s va i q 0b i ,
1 F i F k.
Since 0 g 0b i , the line segment va i coincides with part of vvi starting at v; see Fig. 1. Now take any two vertices, say v1 and v2 , that are connected by an edge in Q. Then v1v2 is also an edge of C. Again Lemma 3.1Žb. implies that v1v2 s a 1 a 2 q b 1b 2 . So the line segment a 1 a 2 Žpossibly a point. is parallel to the edge v1v2 . This means that the triangle convŽv, a 1 , a 2 . is similar to the larger triangle convŽv, v1 , v2 .. Hence
s
,
where
s
gcd Ž a 2 y v . gcd Ž v2 y v .
.
Ž 7.
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IRREDUCIBILITY OF POLYNOMIALS
By Lemma 3.2, the graph of a polytope is connected. Since Eq. Ž7. holds for any two adjacent vertices, we see that
s
gcd Ž a i y v . gcd Ž vi y v .
s t,
1 F i F k,
Ž 8.
where t is a constant 0 F t F 1. This common value t must be a rational number, say mrd, where d G 1, d G m G 0, and gcdŽ m, d . s 1. Then d divides gcdŽvi y v. for 1 F i F k. Suppose that gcdŽv y v1 , v y v2 , . . . , v y vk . s 1. Since gcd Ž v y v1 , v y v2 , . . . , v y vk . s gcd Ž gcd Ž v1 y v . , gcd Ž v2 y v . , . . . , gcd Ž vk y v . . , we see that d must be 1. Hence m s 0 or 1, and so t s 0 or 1. If t s 0 then Ž8. implies that a i s v for 1 F i F k; so A s v4 . If t s 1 then Ž8. implies that a i s vi for 1 F i F k; so A s C and B s 04 . Therefore C is indecomposable. Suppose that gcdŽv y v1 , v y v2 , . . . , v y vk . s d ) 1. Let u i s d1 Žvi y v. for 1 F i F k. Then the u i ’s are integral points in ⺢ n. Define A s conv Ž v, v1 y u 1 , v2 y u 2 , . . . , vk y u k . ,
B s conv Ž 0, u 1 , u 2 , . . . , u k . .
Then it is straightforward to check that A q B s C. Since d ) 1, u i / 0 and vi y u i / v for 1 F i F k. So both A and B have at least two points, and thus C is decomposable. For example, let f be the polynomial 1 q X n q Y m q X n Y m q X i Y j Z k g F w X, Y, Z x where n, m, k ) 0 and i, j G 0. Then the Newton polytope of f is the pyramid with vertices Ž0, 0, 0., Ž n, 0, 0., Ž n, m, 0., Ž0, m, 0., and Ž i, j, k .. If gcdŽ n, m, i, j, k . s 1 then the pyramid is indecomposable and thus f is absolutely irreducible over F. Of course, f remains absolutely irreducible if it is added any number of terms whose exponent vectors lie inside the pyramid. The following corollaries specialize to the simple cases when Q is an integral point, a line segment, or a triangle. COROLLARY 4.3. Let v0 and v1 be two distinct integral points in ⺢ n. Then the line segment v0 v1 is integrally indecomposable iff gcdŽv0 y v1 . s 1. COROLLARY 4.4 ŽOstrowski w34, Theorem IXx.. i kq 1 aX1i1 ⭈⭈⭈ X ki k q bX kq1 ⭈⭈⭈ X ni n g F w X 1 , . . . , X n x ,
A two-term polynomial a, b g F _ 0 4
is absolutely irreducible o¨ er F iff gcdŽ i1 , . . . , i n . s 1, where i1 , . . . , i n4 s 1, . . . , n4 .
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For example, X n q Y m is absolutely irreducible over F iff gcdŽ n, m. s 1; similarly, Y i q X j Z k is absolutely irreducible over F iff gcdŽ i, j, k . s 1. COROLLARY 4.5. Let v0 , v1 , v2 be three integral points in ⺢ n, not all on one line. Then the triangle convŽv0 , v1 , v2 . is integrally indecomposable iff gcd Ž v0 y v1 , v0 y v2 . s 1. COROLLARY 4.6. Let f s aX n q bY m q cX u Y ¨ q Ýc i j X i Y j g F w X, Y x with a, b, c nonzero. Suppose that the Newton polytope of f is the triangle with ¨ ertices Ž n, 0., Ž0, m., and Ž u, ¨ .. If gcdŽ m, n, u, ¨ . s 1 then f is absolutely irreducible o¨ er F. The Newton polytope of the polynomial f in the corollary is the triangle with vertices Ž n, 0., Ž0, m., and Ž u, ¨ . provided that un q m¨ / mn and if c i j / 0 then d Ž mi q nj y mn. G 0, ydŽ ¨ i q Ž n y u. j y ¨ n. G 0, d ŽŽ ¨ y m. i y uj q um. G 0, where d s mu q n¨ y mn. COROLLARY 4.7. Let v0 , v1 , v2 , v3 be four integral points in ⺢ n, not all contained in one plane. Then the tetrahedron convŽv0 , v1 , v2 , v3 . is integrally indecomposable iff gcd Ž v0 y v1 , v0 y v2 , v0 y v3 . s 1. COROLLARY 4.8.
Suppose that the Newton polytope of
f s a1 X l q a2 Y m q a3 Z n q a4 X u Y ¨ Z w q
Ý c i jk X i Y jZ k g F w X , Y , Z x
is the tetrahedron with ¨ ertices Ž l, 0, 0., Ž0, m, 0., Ž0, 0, n., and Ž u, ¨ , w .. If gcd Ž l, m, n, u, ¨ , w . s 1. then f is absolutely irreducible o¨ er F. COROLLARY 4.9. Let Q be any integral polytope in ⺢ n contained in a hyperplane H and let v g ⺢ n be an integral point lying outside of H. If Q has one edge v1v2 such that gcdŽv1 y v2 . s 1 or a ¨ ertex v1 such that gcdŽv y v1 . s 1 then the polytope convŽv, Q . is integrally indecomposable. COROLLARY 4.10. Let f s g Ž X . q hŽ X 1 , . . . , X n . where g g F w X x of degree r and h g F w X 1 , . . . , X n x of total degree m. If gcdŽ r, m. s 1 then f is absolutely irreducible o¨ er F. Proof. By a translation of the variable X, we may assume that the constant of f is nonzero. So the Newton polytope of f is a pyramid with the Newton polytope of h as its base. Since h has total degree m, it has a
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term cX 1k 1 ⭈⭈⭈ X nk n of degree m such that its exponent vector Ž0, k 1 , . . . , k n . s v1 is a vertex of Ph . Since g has degree r, X r has a nonzero coefficient in f and its exponent vector Ž r, 0, . . . , 0. s v is a vertex of the pyramid outside its base. Since gcdŽ m, r . s 1 and k 1 q ⭈⭈⭈ qk n s m, we have gcd Ž v y v1 . s gcd Ž r , k 1 , . . . , k n . s 1. By the above corollary, Pf is indecomposable and so f is absolutely irreducible over F. THEOREM 4.11. Let Q be an indecomposable integral polytope in ⺢ n that is contained in a hyperplane H and has at least two points, and let v g ⺢ n be a point Ž not necessarily integral . lying outside of H. Let S be any set of integral points in the polytope convŽv, Q .. Then the polytope convŽ S, Q . is integrally indecomposable. Proof. Let C s convŽ S, Q . as depicted in Fig. 2. Note that Q s C l H, so Q is a face of C. If C s A q B for some integral polytopes A and B then, by Lemma 3.1, A and B have unique faces A1 and B1 , respectively, such that Q s A1 q B1. Since Q is indecomposable, A1 or B1 must have only one point, say A1. By appropriately shifting A and B, we may assume that A1 s 04 and B1 s Q. We want to show that A s A1 s 04 . This is geometrically ‘‘clear’’ from Fig. 2, as any shift r q Q, r / 0, of Q cannot be contained in the cone convŽv, Q ., not to mention in C. We prove it algebraically. Fix any r g A. Then r q Q : r q B : C. Since Q : H, we have rqQ :Cl ŽrqH..
FIG. 2. C s convŽ S, Q ..
Ž 9.
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Let C1 be the cone generated by v as its vertex and all points in Q. Then C1 = conv Ž v, Q . = C
C1 l H s Q.
and
Ž 10 .
By Ž9., r q Q : C1 l Ž r q H . ,
Ž 11 .
so the latter is nonempty. By Lemma 3.3, there exists a number t G 0 such that
Ž r q H . l C1 s v q t Ž Q y v . .
Ž 12 .
We show that t F 1. Take any a g Q. Then r q a g C1 l Ž r q H .. By Ž12., there exists b g Q such that r q a s v q t Ž b y v.. Also, since r q a g C : convŽv, Q ., there is b1 g Q such that r q a s v q t 1Ž b1 y v. for some 0 F t 1 F 1. Hence t Ž b y v . s t 1 Ž b1 y v . .
Ž 13 .
Assume that H is defined by ␣ g ⺢ n and  g ⺢; i.e., H s x g ⺢ n : ␣ ⭈ x s  4 . Since b, b1 g Q : H, we have ␣ ⭈ b s ␣ ⭈ b1. Equation Ž13. implies that t Ž  y ␣ ⭈ v. s t 1Ž  y ␣ ⭈ v.. Since v f H, ␣ ⭈ v /  . Therefore t s t 1 and so 0 F t F 1. Now Eqs. Ž11. and Ž12. imply that r q Q : v q t Ž Q y v.; i.e., Q : tQ q a,
Ž 14 .
where a s Ž1 y t .v y r g ⺢ n. For any integer k ) 0, applying Ž14. k times yields Q : t k Q q Ž t ky 1 q ⭈⭈⭈ qt q 1 . a.
Ž 15 .
If 0 F t - 1 then Ž15. can be written as Q : t kQ q
tk y 1 ty1
a.
Since Q is bounded, when k ª ⬁, we have Q : 04 q
y1 ty1
as
½
y1 ty1
5
a .
contradicting the assumption that Q has at least two points. Therefore t s 1. Then Ž14. is the same as r q Q : Q.
Ž 16 .
IRREDUCIBILITY OF POLYNOMIALS
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FIG. 3. Indecomposable polytopes.
For any integer k ) 0, applying Ž16. k times yields kr q Q : Q. This is impossible if r / 0, as Q is bounded. Therefore r s 0 and so A s 04 . The theorem is proved. For an example with n s 3, Fig. 3Žd. shows a polytope contained in the tetrahedron with vertices Ž0, 0, 0., Ž n, 0, 0., Ž0, m, 0., and Ž0, 0, u. for some number u. It is indecomposable if gcdŽ n, m. s 1. For the polynomial f s 1 q X n q Y m q X u Z ¨ q Y i Z j q Z w , its Newton polytope is of type Žd. if u - n and i - m. Hence f is absolutely irreducible for any choice of ¨ , w, and j, provided u - n and i - m. The next corollary deals with the case n s 2. COROLLARY 4.12. Let f s aX m q by n q Ýc i j X i Y j g F w X, Y x with a, b nonzero and Ž i, j . different from Ž m, 0., Ž0, n.. Suppose that the Newton polytope of f is contained in the triangle with ¨ ertices Ž m, 0., Ž0, n., and Ž u, ¨ . for some point Ž u, ¨ . g ⺢ 2 . If gcdŽ m, n. s 1 then f is absolutely irreducible o¨ er F. This corollary includes Eisenstein ᎐Dumas and Stepanov᎐Schmidt criteria as special cases, as shown in Figs. 3Ža. ᎐ Žc.. Case Ža. corresponds to the Eisenstein ᎐Dumas criterion. In this case, Pf has only one point on the dotted vertical line Žwhich means that f has degree n in X and the
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SHUHONG GAO
coefficient of X n is a constant in F ., so Pf is contained in a triangle with two vertices on the y-axis. Case Žb. corresponds to the Stepanov᎐Schimdt criterion. In the case Žc., the dotted triangle can be at any position, so Ža. is a special case of it. For an example, the polynomial f s X 2 q y 3 q aX 3 Y q bX 4 y 6 q cX 11 Y 10 has a Newton polytope of type Žc., so it is absolutely irreducible over F for any values of a, b, c. For another example, the polynomial f s X n q Y m q X nq u Y ¨ q X i Y mq j has a Newton polytope of type Žc. whenever gcd Ž n, m . s 1,
j i
-
¨
u
,
where i, u G 1 and j, ¨ G 0. So f is absolutely irreducible. We should emphasize that Theorems 4.2 and 4.11 can be used to build many types of indecomposable polytopes in higher dimensions iteratively from lower dimensions. Thus one can obtain many types of absolutely irreducible polynomials over any given field. Finally, we briefly discuss the relationship of our decomposability to that w13, Chap. 15x. Let P, Q be polytopes in ⺢ n Žnot necessarily of Grunbaun ¨ . integral . Q is said to be homothetic to P if there is a real number t ) 0 and a vector a g ⺢ n such that Q s tP q a s tb q a : b g P 4 . A polytope Q is called homothetically indecomposable if Q s P1 q P2 for any polytopes P1 and P2 then either P1 or P2 is homothetic to Q. Otherwise, Q is called homothetically decomposable. Indecomposable polytopes in this sense have been extensively studied in the literature w8, 16, 26, 27, 40᎐42x. Homothetic decomposability is not comparable with integral decomposability. On the one hand, a triangle is always homothetically indecomposable, but an integral triangle is integrally indecomposable iff the condition in Corollary 4.5 is satisfied. On the other hand, any polygon with more than three edges in the plane is homothetically decomposable, while Fig. 3 shows many integrally indecomposable polygons!
ACKNOWLEDGMENTS The author thanks Joel Brawley, Erich Kaltofen, Jenny Key, Hendrik Lenstra, Jr., and Daqing Wan for their useful comments on an earlier version of the paper and for bringing some of the references to his attention. Thanks is also due to Joe Mott for sending a copy of his paper to the author. A preliminary version of the paper was presented at an Oberwolfach Workshop on Designs and Codes, March 15᎐21, 1998, Germany, and at a Dagstuhl-Seminar on Algorithms and Number Theory, October 26᎐30, 1998, Germany.
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Absolute Irreducibility of Polynomials via Newton Polytopes Shuhong Gao Department of Mathematical Sciences, Clemson Uni¨ ersity, Clemson, South Carolina 29634 E-mail: [email protected] Communicated by Walter Feit Received July 12, 1999
A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping a certain collection of them nonzero. 䊚 2001 Academic Press Key Words: fields; multivariable polynomials; absolute irreducibility; Newton polytopes; Newton polygons; indecomposable polytopes; Eisenstein criterion; Minkowski sums.
1. INTRODUCTION It is well known that Eisenstein’s criterion gives a simple condition for a polynomial to be irreducible. Over the years this criterion has witnessed many variations and generalizations using Newton polygons, prime ideals, and valuations; see for example w3, 25, 28, 38x. We examine the Newton polygon method and generalize it through Newton polytopes associated with multivariable polynomials. This leads us to a more general geometric criterion for absolute irreducibility of multivariable polynomials. Since the Newton polygon of a polynomial is only a small fraction of its Newton polytope, our method is much more powerful. Absolute irreducibility of polynomials is crucial in many applications including but not limited to finite geometry w14x, combinatorics w47x, algebraic geometric codes w45x 501 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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permutation polynomials w23x, and function field sieve w1x. We present many infinite families of absolutely irreducible polynomials over an arbitrary field. These polynomials remain absolutely irreducible even if their coefficients are modified arbitrarily but with a certain collection of them nonzero. As in many standard algebra textbooks, Eisenstein’s criterion w5x is described as follows. EISENSTEIN’S CRITERION. Let R be a unique factorization domain and let f s f 0 q f 1 X q ⭈⭈⭈ qf n X n g Rw X x. If there is a prime p g R such that all the coefficients except f n of f are di¨ isible by p, but f 0 is not di¨ isible by p 2 , then f is irreducible o¨ er the fraction field of R. Several people ŽDumas w4x, Kurschak w22x, Ore w30᎐32x, Rella w35x. have generalized this criterion by using Newton polygons. Assume that f 0 f n / 0. One can construct a polygon in the Euclidean plane as follows. Suppose that the coefficient f i is divisible by p a i but not any higher power, where a i G 0 and a i is undefined if f i s 0. Plot the points Ž0, a0 ., Ž1, a1 ., . . . , Ž n, a n . in the Euclidean plane and form the lower convex hull of these points. This results in a sequence of line segments in the first quadrant of the plane, called the Newton polygon of f Žwith respect to the prime p .. Dumas w4x determines the degrees of all the possible nontrivial factors of f in terms of the widths of the line segments on the Newton polygon of f. Consequently a simple criterion for the irreducibility of f is established. EISENSTEIN ᎐DUMAS CRITERION. Let R be a unique factorization domain and let f s f 0 q f 1 X q ⭈⭈⭈ qf n X n g Rw X x with f 0 f n / 0. Assume that f is primiti¨ e; i.e., f 0 , . . . , f n ha¨ e no nontri¨ ial common factor in R. If the Newton polygon of f with respect to some prime p g R consists of the only line segment from Ž0, m. to Ž n, 0. and if gcdŽ n, m. s 1 then f is irreducible in Rw X x. The condition on the Newton polygon means that a i G Ž n y i . mrn for 0 F i F n where p a i exactly divides f i . When m s a0 s 1, this condition is the same as in Eisenstein’s criterion. Hence the Eisenstein ᎐Dumas criterion generalizes that of Eisenstein. The Eisenstein ᎐Dumas criterion was originally proved for integer coefficients. Later it was generalized to local fields or any field with valuations w3, 20, 25x. We are interested in the case when R is a polynomial ring over a field. Let F be a field and let R s F w Y x where Y is a new variable. Then Y is a prime in R and the Eisenstein ᎐Dumas criterion can be applied in Rw X x ( F w X, Y x. We restate the criterion as follows. EISENSTEIN ᎐DUMAS CRITERION Ža special case.. Let F be any field and let f s f 0 Ž Y . q f 1Ž Y . X q ⭈⭈⭈ qf nŽ Y . X n g F w X, Y x. Assume that f 0 Ž Y . / 0
IRREDUCIBILITY OF POLYNOMIALS
503
and f nŽ Y . is a nonzero constant in F. If the Newton polygon of f Ž with respect to Y . has only one line segment from Ž0, m. to Ž n, 0. and gcdŽ n, m. s 1, then f is Ž absolutely . irreducible o¨ er F. A polynomial over a field F is called absolutely irreducible if it remains irreducible over every algebraic extension of F. The same proof for the irreducibility of f under the Eisenstein ᎐Dumas condition also shows that f is absolutely irreducible. The above criterion was also discovered by Wan w48x. In w36, Theorem 1B, p. 92x, Schmidt describes another method for constructing absolutely irreducible polynomials which he attributes to Stepanov w43, 44x. This method can also be interpreted as a polygon method. Let f s f 0 Ž Y . q f 1Ž Y . X q ⭈⭈⭈ qf nŽ Y . X n g F w X, Y x. The upper Newton polygon of f with respect to Y is defined to be the upper convex hull of the points Ž0, a0 ., Ž1, a1 ., . . . , Ž n, a n . where a i is the degree of f i Ž Y . in Y and a i is not defined if f i Ž Y . s 0. STEPANOV ᎐SCHMIDT CRITERION. Let F be a field and let f g F w X, Y x with degree n in X. If the upper Newton polygon of f with respect to Y has only one line segment from Ž0, m. to Ž n, 0. and gcdŽ n, m. s 1, then f is absolutely irreducible o¨ er F. Note that the Eisenstein ᎐Dumas and Stepanov᎐Schmidt criteria read exactly the same except that they exploit ‘‘different parts’’ of the polynomials. This leads us to consider the convex hull of the exponent vectors Ž i, j . of all the nonzero terms cX i Y j of a polynomial f and call the resulted convex set the Newton polytope of f. Its boundary gives us a ‘‘whole’’ polygon that contains both the lower and upper polygons used above. This concept of Newton polytopes associated with polynomials is due to Ostrowski Ž1921. and is similarly defined for any multivariable polynomials. Ostrowski realizes that the factorization of polynomials implies the decomposition of polytopes in the sense of the Minkowski sum. In the 1970s, Ostrowski wrote two papers w33, 34x dealing with term ordering and irreducibility of multivariable polynomials. His irreducibility criteria are, however, based mainly on algebraic techniques Žsuch as algebraic independence. and Puiseux developments. We show that Newton polytopes carry a lot of information about the irreducibility of polynomials. Indeed, the Eisenstein ᎐Dumas and Stepanov᎐Schmidt criteria are just very special cases of our results. More precisely, we study the irreducibility of multivariable polynomials through the decomposability of their Newton polytopes. Our main contribution is in the construction of indecomposable polytopes and thus we give many classes of absolutely irreducible polynomials over an arbitrary field.
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To get a glimpse of our results, we give two examples here; more general results can be found in Section 4. EXAMPLE 1. Let F be any field and let f s aX n q bY m q cX u Y ¨ q
Ý ci j X i Y j g F w X , Y x
with a, b, c nonzero. Suppose that the Newton polytope of f is the triangle with vertices Ž n, 0., Ž0, m., and Ž u, ¨ .. If gcdŽ m, n, u, ¨ . s 1 then f is absolutely irreducible over F. EXAMPLE 2. Suppose that the Newton polytope of f s a1 X l q a2 Y m q a3 Z n q a4 X u Y ¨ Z w q
Ý c i jk X i Y jZ k g F w X , Y , Z x
is the tetrahedron with vertices Ž l, 0, 0., Ž0, m, 0., Ž0, 0, n., and Ž u, ¨ , w .. Then f is absolutely irreducible over F, provided gcdŽ l, m, n, u, ¨ , w . s 1. Our paper is organized as follows. In the next section, we define the decomposability of polytopes with respect to Minkowski sums and discuss its relation with the factorization of polynomials. A general irreducibility criterion is established. Note our concept of decomposability of polytopes is incompatible with that in the literature, say in Grunbaun’s book w13, ¨ x Chap. 15 . In Section 3, we collect properties of Minkowski sums of polytopes, particularly on the decomposition of faces of polytopes. In Section 4, we give two general constructions of indecomposable polytopes and thus give many simple and explicit criteria for absolute irreducibility of multivariable polynomials. Many infinite families of absolutely irreducible polynomials are described. Related Works We should mention that Lipkovski w24x associates a polynomial with a polyhedron Žunbounded., called its Newton polyhedron, which is a direct analogue of Newton polygon in higher dimension. Indecomposability of its Newton polyhedron implies the analytic irreducibility of a polynomial at the origin Ži.e., irreducibility in the formal power series ring.. To that extent, Lipkovski’s method is local while our polytope method is somewhat global. Lipkovski discusses indecomposability of polyhedra and gives several constructions of indecomposable polyhedra. Filaseta w7x uses the Newton polygon method to decide irreducibility of Bessel polynomials. Wan w49x uses the Newton polygon to study zeta functions and L functions. Gao and Shokrollahi w10x use the Newton polygon method to compute roots of polynomials over unction fields of curves. For a survey of the Newton polygon method, see Mott w28x.
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There are several other methods in the literature for proving absolute irreducibility. One is using singularity analysis: if a polynomial defines a smooth hypersurface then it is absolutely irreducible. This method is often used in algebraic geometry. Another method is presented by Janwa et al. w15x using Bezout’s theorem on intersection multiplicity of curves. Noether’s irreducibility forms w29x give yet another powerful method for proving irreducibility of polynomials. Noether’s forms are carefully analysed by Schmidt w36x and greatly improved by Kaltofen w19x. For efficient algorithms for testing irreducibility of multivariable polynomials, see Gao w9x, von zur Gathen w11x, and Kaltofen w17, 18x. We should also mention that Newton polytopes have been used extensively to study toric ideals and solutions of systems of Žmultivariable. polynomial equations; see Gel’fand et al. w12x, Khovanskiı˘ w21x, Sturmfels w46x, and the references therein.
2. POLYTOPE METHOD We assume that the reader is familiar with the basic properties of w13x, Webster w50x, and Ziegler w52x. polytopes; see Ewald w6x, Grunbaun ¨ For the convenience of the reader, we review some basic concepts that will be needed in the sequel. Let ⺢ be the set of real numbers and let n be a positive integer. A subset S : ⺢ n is called convex if, for any two points a, b g S, the line segment from a to b is also contained in S; that is, a q Ž b y a . s Ž 1 y . a q b g S,
᭙0 F F 1.
For any subset S : ⺢ n, convŽ S . denotes the smallest convex set in ⺢ n that contains S. It is straightforward to check that conv Ž S . s
½
t
t
Ý i ai : ai g S, i G 0,
Ý i s 1
is1
is1
5
.
When S s a1 , . . . , a k 4 is a finite set, denote convŽ S . by convŽ a1 , . . . , a k ., which is called the convex hull of a1 , . . . , a k . The convex hull of finitely many points is called a polytope. A point of a polytope is called a vertex if it is not on the line segment of any other two points of the polytope. It is well known that a polytope is always the convex hull of its vertices. We consider polynomials with n variables X 1 , X 2 , . . . , X n . Let F be any field and let f s Ý f i1 i 2 ⭈ ⭈ ⭈ i nX 1i1 X 2i 2 ⭈⭈⭈ X ni n g F w X 1 , X 2 , . . . , X n x. An exponent vector Ž i1 , i 2 , . . . , i n . of f can be considered as a point in ⺢ n. The
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Newton polytope of f, denoted by Pf , is defined to be the convex hull in ⺢ n of all the points Ž i1 , i 2 , . . . , i n . with f i1 i 2 ⭈ ⭈ ⭈ i n / 0. For two sets A and B in ⺢ n, define A q B s a q b : a g A, b g B4 , which is called the Minkowski sum of A and B. LEMMA 2.1 ŽOstrowski w33x.. gh. Then Pf s Pg q Ph .
Let f, g, h g F w X 1 , X 2 , . . . , X n x with f s
Proof. This result is well known in the literature. For the sake of completeness, we give a simple proof here. By multiplication of polynomials, it is obvious that Pf : Pg q Ph . To prove the other inclusion, let v be any vertex of Pg q Ph . We show that there are unique points v1 g Pg and v2 g Ph such that v s v1 q v2 . Since v g Pg q Ph , the existence is no problem. Suppose that there is another pair v1X g Pg and v2X g Ph such that v s v1 q v2 s v1X q v2X .
Ž 1.
Then vs
1 2
Ž v1 q v2X . q
1 2
Ž v1X q v2 . .
Since v1 q v2X , v1X q v2 g Pg q Ph and v is a vertex of Pg q Ph , one must have v1 q v2X s v1X q v2 .
Ž 2.
Subtracting Ž2. from Ž1. yields v2 y v2X s v2X y v2 ,
i.e., 2 Ž v2 y v2X . s 0.
Hence v2 s v2X and v1 s v1X . Since v is a vertex of Pg q Ph , v1 and v2 must be vertices of Pg and Ph , respectively. There is a unique term in the expansion of g ⭈ h that has v as its exponent vector. Hence v g Pf . This proves that all the vertices of Pg q Ph are in Pf . Consequently, Pf = Pg q Ph as a polytope is the convex hull of its vertices. A point in ⺢ n is called integral if its coordinates are integers. A polytope in ⺢ n is called integral if all of its vertices are integral. Certainly, Newton polytopes of polynomials are integral. An integral polytope C is called integrally decomposable if there exist integral polytopes A and B such that C s A q B where both A and B have at least two points. Otherwise, we say that C is integrally indecomposable. Note that our concept of indecomposability is different from that in Grunbaun’s book ¨ w13, Chap. 15x; see the comments at the end of the paper. Since we will not encounter any other type of decomposability in this paper, the word ‘‘integrally’’ will be freely omitted in the sequel.
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IRREDUCIBILITY CRITERION. Let F be any field and let f g F w X 1 , X 2 , . . . , X n x be a nonzero polynomial not di¨ isible by any X i . If the Newton polytope of f is integrally indecomposable then f is absolutely irreducible o¨ er F. Proof. First note that f has no factor with only one term. Suppose that f factors nontrivially over some algebraic extension of F, say f s gh, where both g and h have at least two nonzero terms. Then the Newton polytopes of g and h have at least two points. By Lemma 2.1, Pf s Pg q Ph , contradicting our assumption that Pf is integrally indecomposable. When Pf is decomposable, f can be either reducible or irreducible. For example, if f s 1 q Y q XY q X 2 q Y 2 then Pf is decomposable Žequal to the sum of the triangle Ž0, 0. ᎐ Ž1, 0. ᎐ Ž0, 1. with itself.. Over a field F of characteristic different from two, it can be verified directly that f is absolutely irreducible. Over a field F of characteristic two, however, we have f s Ž1 q X q Y . Ž1 q X q 2Y . , where is an element of order 3 Žso f is irreducible over F if f F .. It remains to show that indecomposable Newton polytopes exist; thus absolutely irreducible polynomials can be constructed via the irreducibility criterion above. This will be our main focus in Section 4. We conclude this section with some comments. Remarks. Ž1. One can change the coefficients of a polynomial f arbitrarily and its Newton polytope will remain the same provided the coefficients of all the terms of f that correspond to vertices are nonzero. If the Newton polytope of f is indecomposable then f will remain absolutely irreducible when its coefficients are modified arbitrarily but with those of vertices nonzero. This gives great freedom in choosing suitable polynomials in applications. For all the examples in the sequel, we often give polynomials with coefficients fixed at 1, but one may change the coefficients to any nonzero elements in the ground field. Ž2. In w33x, Ostrowski uses the term ‘‘baric polyhedron’’ in place of the ‘‘Newton polytope’’ of a polynomial. Lemma 2.1 is proved for more general polynomials called algebraic polynomials where the exponents of variables may be rational numbers. Ostrowski mentions that he gave a talk at a German Mathematical Society meeting in 1921 about the baric polyhedron and its applications to the irreducibility problem. In his sequel paper w34x, Ostrowski discusses irreducibility of polynomials in details. It is surprising, however, that the concept of decomposability of polytopes does not arise there. Ostrowski develops irreducibility criteria using mainly
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algebraic tools Žsuch as algebraic independence and conjugates. and the Puiseux developments. Ž3. For the case of polynomials with three variables, the concept of the Newton polytope also appears in Shanok’s paper w39x. Shanok develops irreducibility criteria for polynomials by projecting a Newton polytope from ⺢ 3 into planes. Ž4. As mentioned in the introduction, Lipkovski w24x develops an analogue of the Newton polygon method in higher dimension for formal power series. Lipkovski associates a power series f of n variables with a Newton polyhedron Pf q ⺢ 0n , where Pf is defined similarly to polynomials and ⺢ 0 is the set of nonnegative real numbers. A Newton polyhedron is unbounded and, when n s 2, its finite edges form the Newton polygon. Lipkovski defines a similar concept of decomposability of Newton polyhedra and gives several constructions of indecomposable Newton polyhedra. For a polynomial of two variables, the only indecomposable Newton polyhedron is that corresponding to the Newton polygon as described in Eisenstein ᎐Dumas criteria. We will see, however, that there are many indecomposable polytopes in dimension two or higher.
3. SOME PROPERTIES OF POLYTOPES To further discuss the decomposability of polytopes, we need more properties about Minkowski sums of polytopes, particularly on the decomposition of their faces. Minkowski sums of convex sets have been extensively studied in the literature; see for example Schneider w37x. Let P be a polytope in ⺢ n. A face of P is by definition the intersection of P with a supporting hyperplane to P. In other words, a face of P is the set of all the points in P that maximize some linear function. A vertex is a just face of dimension 0. A face of dimension 1 is a line segment, called an edge of P. A face of dimension 1 less than that of P is called facet of P. The next result describes how faces decompose in a Minkowski sum of w13, polytopes; for its proof, see Ewald w6, Theorem 1.5x, Grunbaun ¨ Theorem 1, p. 317x, or Schneider w37, Theorem 1.7.5x. LEMMA 3.1.
Let A and B be polytopes in ⺢ n and let C s A q B.
Ža. Each face of C is a Minkowski sum of unique faces of A and B. Žb. Let C1 be any face of C and let c1 , c 2 , . . . , c k all of its ¨ ertices. Suppose that c i s a i q bi where a i g A and bi g B for 1 F i F k. Let A1 s conv Ž a1 , a2 , . . . , a k . ,
B1 s conv Ž b1 , b 2 , . . . , bk . .
Then A1 and B1 are faces of A and B, respecti¨ ely, and C1 s A1 q B1.
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Note that part Žb. is the constructive version of part Ža. and it says that the decomposition of all the faces is determined by the decomposition of vertices alone. This is extremely useful in applications. Part Ža. can be strengthened as follows. Let C1 be any face of C. Suppose that A1 and B1 are any convex subsets of A and B, respectively, such that C1 s A1 q B1. Then A1 and B1 must be faces of A and B, respectively. The proof is a little subtle and will be given elsewhere. A polytope is associated naturally with a graph consisting of its vertices and edges. LEMMA 3.2 ŽBalinski w2x.. The graph of a polytope of dimension n is n-connected. A convex cone with a vertex v is defined to be a convex set S in ⺢ n such that v is an extreme point of S and, for any a g S, v q Ž a y v. g S for all real numbers G 0. The next result must be known in the literature, but we could not find a convenient reference, so a proof is included. LEMMA 3.3. Let C be a con¨ ex cone with ¨ ertex v and let H be a hyperplane in ⺢ n with v f H. Suppose that Q s C l H is nonempty and bounded. Then, for any r g ⺢ n, either C l Ž r q H . is empty or there exists a real number t G 0 such that C l Ž r q H . s v q t Ž Q y v. s v q t Ž a y v. : a g Q4 . Proof. Choose ␣ g ⺢ n and  g ⺢ such that H s x g ⺢n : ␣ ⭈ x s  4
and
␣ ⭈ v ) .
We show that for every point a g C with a / v,
␣ ⭈ a - ␣ ⭈ v.
Ž 3.
Suppose on the contrary that ␣ ⭈ a0 G ␣ ⭈ v for some a0 g C. Let b g Q s C l H be any fixed point. Then
␣ ⭈ b s  - ␣ ⭈ v F ␣ ⭈ a0 . Let a1 s 1 a0 q Ž1 y 1 . b where 1 s Ž ␣ ⭈ v y  .rŽ ␣ ⭈ a0 y  . ) 0. Since 1 F 1 and C is convex, we have a1 g C and
␣ ⭈ a1 s ␣ ⭈ v.
Ž 4.
For any t G 0, b q t Ž a1 y v . s v q Ž t q 1 .
žž
t tq1
a1 q
1 tq1
b yv
/ /
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belongs to C, as a1 , b g C and C is a convex cone with vertex v. By Ž4., ␣ ⭈ Ž b q t Ž a1 y v.. s ␣ ⭈ b s  . Hence b q t Ž a1 y v. g H and b q t Ž a1 y v. g C l H s Q for all t G 0, contradicting the boundedness of Q Žnote that a1 / v.. Therefore Ž3. holds. For any r g ⺢ n and any a g C with a / v, consider the intersection of the ray
v q Ž a y v . : G 04
Ž 5.
with the hyperplane r q H s r q x g ⺢ n : ␣ ⭈ x s  4 s x g ⺢ n : ␣ ⭈ x s ␣ ⭈ r q  4 . Ž 6. Note that ␣ ⭈ Žv q Ž a y v.. s ␣ ⭈ r q  implies that s Ž ␣ ⭈ v y  y ␣ ⭈ r .rŽ ␣ ⭈ v y ␣ ⭈ a.. Since ␣ ⭈ v y ␣ ⭈ a ) 0 by Ž3., G 0 iff ␣ ⭈ r F ␣ ⭈ v y  . When this condition is satisfied, r q H intersects every ray Ž5. at a unique point determined by above. For r s 0, since ␣ ⭈ v y  ) 0 s ␣ ⭈ r, each ray Ž5. intersects H, and thus Q, at a unique point. Hence we may index all the rays Ž5. by a g Q. Now suppose that ␣ ⭈ r F ␣ ⭈ v y  . Then for each a g Q, the ray Ž5. intersects r q H at the point b s v q 0 Ž a y v. where
0 s
␣⭈vyy␣⭈r ␣⭈vy
.
Therefore the lemma holds with t s 0 . 4. INDECOMPOSABLE POLYTOPES AND ABSOLUTELY IRREDUCIBLE POLYNOMIALS We now proceed to construct indecomposable polytopes in ⺢ n. Each type of indecomposable polytope gives us a family of absolutely irreducible polynomials. When a polytope has only one point we say that it is trivial. We examine several types of simple nontrivial polytopes such as line segments, triangles, tetrahedrons, pyramids, etc. We show how to construct indecomposable polytopes from a given polytope. We need more terminology. A line segment convŽv1 , v2 . is simply denoted by v1v2 . For an integral point or vector v s Ž a1 , . . . , a n ., we write gcdŽv. to mean gcdŽ a1 , . . . , a n ., i.e., the greatest common divisor of the components in v. Similarly, for several points v1 , . . . , vk , gcdŽv1 , . . . , vk . means that gcd of all the components in v1 , . . . , vk together. For example, if v1 s Ž n, 0., v 2 s Ž0, m ., and v 3 s Ž u, u ., then gcdŽv1 , v 2 , v 3 . s gcdŽ n, 0, 0, m, u, u. s gcdŽ n, m, u.. For any two integral vectors v1 and v2 , we have gcdŽv1 , v2 . s gcdŽv1 , v2 y t v1 . for any integer t.
IRREDUCIBILITY OF POLYNOMIALS
511
LEMMA 4.1. Let v0 and v1 be two distinct integral points in ⺢ n. Then the number of integral points on the line segment v0 v1 , including v0 and v1 , is equal to gcdŽv0 y v1 . q 1. Further, if v2 is any integral point on v0 v1 , then
s
gcd Ž v2 y v0 . gcd Ž v1 y v0 .
,
where
0 F t F 1.
Since v0 is integral, v is integral iff t Žv1 y v0 . is integral. But the components of v1 y v0 are all integers, so t must be rational. Let ts
i k
,
for some 0 F i - k with gcd Ž k , i . s 1.
Then t Žv1 y v0 . is integral iff k < gcdŽv1 y v0 .. Hence if v is an integral point different from v0 and v1 , then t must be of the form ts
i d
,
0 - i - d,
where d s gcdŽv1 y v0 . G 1. The number of choices for i is d y 1. So the total number of integral points v on v0 v1 is d y 1 q 2 s d q 1. Suppose v2 s v0 q ird Žv1 y v0 . is any integral point on v0 v1 with 0 F i F d where d s gcdŽv1 y v0 .. Note that Žv1 y v0 .rd is integral and gcdŽŽv1 y v0 .rd . s 1. Hence gcdŽv2 y v0 . s gcdŽ i ⭈ Žv1 y v0 .rd . s i. Also
Therefore the equation in the lemma holds. THEOREM 4.2. Let Q be any integral polytope in ⺢ n contained in a hyperplane H and let v g ⺢ n be an integral point lying outside of H. Suppose that v1 , v2 , . . . , vk are all the ¨ ertices of Q. Then the polytope convŽv, Q . is integrally indecomposable iff gcd Ž v y v1 , v y v2 , . . . , v y vk . s 1. Proof. Let C s convŽv, Q . as depicted in Fig. 1. Suppose that C s A q B for some integral polytopes A and B in ⺢ n. By appropriately shifting A and B, we may assume that v g A and 0 g B. Note that v, v1 , . . . , vk are all
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FIG. 1. Indecomposable pyramid.
the vertices of C, and vv1 , . . . , vvk are edges of C. By Lemma 3.1, there are unique vertices a i g A and b i g B such that vi s a i q b i ,
1 F i F k,
and vvi s va i q 0b i ,
1 F i F k.
Since 0 g 0b i , the line segment va i coincides with part of vvi starting at v; see Fig. 1. Now take any two vertices, say v1 and v2 , that are connected by an edge in Q. Then v1v2 is also an edge of C. Again Lemma 3.1Žb. implies that v1v2 s a 1 a 2 q b 1b 2 . So the line segment a 1 a 2 Žpossibly a point. is parallel to the edge v1v2 . This means that the triangle convŽv, a 1 , a 2 . is similar to the larger triangle convŽv, v1 , v2 .. Hence
s
,
where
s
gcd Ž a 2 y v . gcd Ž v2 y v .
.
Ž 7.
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IRREDUCIBILITY OF POLYNOMIALS
By Lemma 3.2, the graph of a polytope is connected. Since Eq. Ž7. holds for any two adjacent vertices, we see that
s
gcd Ž a i y v . gcd Ž vi y v .
s t,
1 F i F k,
Ž 8.
where t is a constant 0 F t F 1. This common value t must be a rational number, say mrd, where d G 1, d G m G 0, and gcdŽ m, d . s 1. Then d divides gcdŽvi y v. for 1 F i F k. Suppose that gcdŽv y v1 , v y v2 , . . . , v y vk . s 1. Since gcd Ž v y v1 , v y v2 , . . . , v y vk . s gcd Ž gcd Ž v1 y v . , gcd Ž v2 y v . , . . . , gcd Ž vk y v . . , we see that d must be 1. Hence m s 0 or 1, and so t s 0 or 1. If t s 0 then Ž8. implies that a i s v for 1 F i F k; so A s v4 . If t s 1 then Ž8. implies that a i s vi for 1 F i F k; so A s C and B s 04 . Therefore C is indecomposable. Suppose that gcdŽv y v1 , v y v2 , . . . , v y vk . s d ) 1. Let u i s d1 Žvi y v. for 1 F i F k. Then the u i ’s are integral points in ⺢ n. Define A s conv Ž v, v1 y u 1 , v2 y u 2 , . . . , vk y u k . ,
B s conv Ž 0, u 1 , u 2 , . . . , u k . .
Then it is straightforward to check that A q B s C. Since d ) 1, u i / 0 and vi y u i / v for 1 F i F k. So both A and B have at least two points, and thus C is decomposable. For example, let f be the polynomial 1 q X n q Y m q X n Y m q X i Y j Z k g F w X, Y, Z x where n, m, k ) 0 and i, j G 0. Then the Newton polytope of f is the pyramid with vertices Ž0, 0, 0., Ž n, 0, 0., Ž n, m, 0., Ž0, m, 0., and Ž i, j, k .. If gcdŽ n, m, i, j, k . s 1 then the pyramid is indecomposable and thus f is absolutely irreducible over F. Of course, f remains absolutely irreducible if it is added any number of terms whose exponent vectors lie inside the pyramid. The following corollaries specialize to the simple cases when Q is an integral point, a line segment, or a triangle. COROLLARY 4.3. Let v0 and v1 be two distinct integral points in ⺢ n. Then the line segment v0 v1 is integrally indecomposable iff gcdŽv0 y v1 . s 1. COROLLARY 4.4 ŽOstrowski w34, Theorem IXx.. i kq 1 aX1i1 ⭈⭈⭈ X ki k q bX kq1 ⭈⭈⭈ X ni n g F w X 1 , . . . , X n x ,
A two-term polynomial a, b g F _ 0 4
is absolutely irreducible o¨ er F iff gcdŽ i1 , . . . , i n . s 1, where i1 , . . . , i n4 s 1, . . . , n4 .
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For example, X n q Y m is absolutely irreducible over F iff gcdŽ n, m. s 1; similarly, Y i q X j Z k is absolutely irreducible over F iff gcdŽ i, j, k . s 1. COROLLARY 4.5. Let v0 , v1 , v2 be three integral points in ⺢ n, not all on one line. Then the triangle convŽv0 , v1 , v2 . is integrally indecomposable iff gcd Ž v0 y v1 , v0 y v2 . s 1. COROLLARY 4.6. Let f s aX n q bY m q cX u Y ¨ q Ýc i j X i Y j g F w X, Y x with a, b, c nonzero. Suppose that the Newton polytope of f is the triangle with ¨ ertices Ž n, 0., Ž0, m., and Ž u, ¨ .. If gcdŽ m, n, u, ¨ . s 1 then f is absolutely irreducible o¨ er F. The Newton polytope of the polynomial f in the corollary is the triangle with vertices Ž n, 0., Ž0, m., and Ž u, ¨ . provided that un q m¨ / mn and if c i j / 0 then d Ž mi q nj y mn. G 0, ydŽ ¨ i q Ž n y u. j y ¨ n. G 0, d ŽŽ ¨ y m. i y uj q um. G 0, where d s mu q n¨ y mn. COROLLARY 4.7. Let v0 , v1 , v2 , v3 be four integral points in ⺢ n, not all contained in one plane. Then the tetrahedron convŽv0 , v1 , v2 , v3 . is integrally indecomposable iff gcd Ž v0 y v1 , v0 y v2 , v0 y v3 . s 1. COROLLARY 4.8.
Suppose that the Newton polytope of
f s a1 X l q a2 Y m q a3 Z n q a4 X u Y ¨ Z w q
Ý c i jk X i Y jZ k g F w X , Y , Z x
is the tetrahedron with ¨ ertices Ž l, 0, 0., Ž0, m, 0., Ž0, 0, n., and Ž u, ¨ , w .. If gcd Ž l, m, n, u, ¨ , w . s 1. then f is absolutely irreducible o¨ er F. COROLLARY 4.9. Let Q be any integral polytope in ⺢ n contained in a hyperplane H and let v g ⺢ n be an integral point lying outside of H. If Q has one edge v1v2 such that gcdŽv1 y v2 . s 1 or a ¨ ertex v1 such that gcdŽv y v1 . s 1 then the polytope convŽv, Q . is integrally indecomposable. COROLLARY 4.10. Let f s g Ž X . q hŽ X 1 , . . . , X n . where g g F w X x of degree r and h g F w X 1 , . . . , X n x of total degree m. If gcdŽ r, m. s 1 then f is absolutely irreducible o¨ er F. Proof. By a translation of the variable X, we may assume that the constant of f is nonzero. So the Newton polytope of f is a pyramid with the Newton polytope of h as its base. Since h has total degree m, it has a
IRREDUCIBILITY OF POLYNOMIALS
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term cX 1k 1 ⭈⭈⭈ X nk n of degree m such that its exponent vector Ž0, k 1 , . . . , k n . s v1 is a vertex of Ph . Since g has degree r, X r has a nonzero coefficient in f and its exponent vector Ž r, 0, . . . , 0. s v is a vertex of the pyramid outside its base. Since gcdŽ m, r . s 1 and k 1 q ⭈⭈⭈ qk n s m, we have gcd Ž v y v1 . s gcd Ž r , k 1 , . . . , k n . s 1. By the above corollary, Pf is indecomposable and so f is absolutely irreducible over F. THEOREM 4.11. Let Q be an indecomposable integral polytope in ⺢ n that is contained in a hyperplane H and has at least two points, and let v g ⺢ n be a point Ž not necessarily integral . lying outside of H. Let S be any set of integral points in the polytope convŽv, Q .. Then the polytope convŽ S, Q . is integrally indecomposable. Proof. Let C s convŽ S, Q . as depicted in Fig. 2. Note that Q s C l H, so Q is a face of C. If C s A q B for some integral polytopes A and B then, by Lemma 3.1, A and B have unique faces A1 and B1 , respectively, such that Q s A1 q B1. Since Q is indecomposable, A1 or B1 must have only one point, say A1. By appropriately shifting A and B, we may assume that A1 s 04 and B1 s Q. We want to show that A s A1 s 04 . This is geometrically ‘‘clear’’ from Fig. 2, as any shift r q Q, r / 0, of Q cannot be contained in the cone convŽv, Q ., not to mention in C. We prove it algebraically. Fix any r g A. Then r q Q : r q B : C. Since Q : H, we have rqQ :Cl ŽrqH..
FIG. 2. C s convŽ S, Q ..
Ž 9.
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Let C1 be the cone generated by v as its vertex and all points in Q. Then C1 = conv Ž v, Q . = C
C1 l H s Q.
and
Ž 10 .
By Ž9., r q Q : C1 l Ž r q H . ,
Ž 11 .
so the latter is nonempty. By Lemma 3.3, there exists a number t G 0 such that
Ž r q H . l C1 s v q t Ž Q y v . .
Ž 12 .
We show that t F 1. Take any a g Q. Then r q a g C1 l Ž r q H .. By Ž12., there exists b g Q such that r q a s v q t Ž b y v.. Also, since r q a g C : convŽv, Q ., there is b1 g Q such that r q a s v q t 1Ž b1 y v. for some 0 F t 1 F 1. Hence t Ž b y v . s t 1 Ž b1 y v . .
Ž 13 .
Assume that H is defined by ␣ g ⺢ n and  g ⺢; i.e., H s x g ⺢ n : ␣ ⭈ x s  4 . Since b, b1 g Q : H, we have ␣ ⭈ b s ␣ ⭈ b1. Equation Ž13. implies that t Ž  y ␣ ⭈ v. s t 1Ž  y ␣ ⭈ v.. Since v f H, ␣ ⭈ v /  . Therefore t s t 1 and so 0 F t F 1. Now Eqs. Ž11. and Ž12. imply that r q Q : v q t Ž Q y v.; i.e., Q : tQ q a,
Ž 14 .
where a s Ž1 y t .v y r g ⺢ n. For any integer k ) 0, applying Ž14. k times yields Q : t k Q q Ž t ky 1 q ⭈⭈⭈ qt q 1 . a.
Ž 15 .
If 0 F t - 1 then Ž15. can be written as Q : t kQ q
tk y 1 ty1
a.
Since Q is bounded, when k ª ⬁, we have Q : 04 q
y1 ty1
as
½
y1 ty1
5
a .
contradicting the assumption that Q has at least two points. Therefore t s 1. Then Ž14. is the same as r q Q : Q.
Ž 16 .
IRREDUCIBILITY OF POLYNOMIALS
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FIG. 3. Indecomposable polytopes.
For any integer k ) 0, applying Ž16. k times yields kr q Q : Q. This is impossible if r / 0, as Q is bounded. Therefore r s 0 and so A s 04 . The theorem is proved. For an example with n s 3, Fig. 3Žd. shows a polytope contained in the tetrahedron with vertices Ž0, 0, 0., Ž n, 0, 0., Ž0, m, 0., and Ž0, 0, u. for some number u. It is indecomposable if gcdŽ n, m. s 1. For the polynomial f s 1 q X n q Y m q X u Z ¨ q Y i Z j q Z w , its Newton polytope is of type Žd. if u - n and i - m. Hence f is absolutely irreducible for any choice of ¨ , w, and j, provided u - n and i - m. The next corollary deals with the case n s 2. COROLLARY 4.12. Let f s aX m q by n q Ýc i j X i Y j g F w X, Y x with a, b nonzero and Ž i, j . different from Ž m, 0., Ž0, n.. Suppose that the Newton polytope of f is contained in the triangle with ¨ ertices Ž m, 0., Ž0, n., and Ž u, ¨ . for some point Ž u, ¨ . g ⺢ 2 . If gcdŽ m, n. s 1 then f is absolutely irreducible o¨ er F. This corollary includes Eisenstein ᎐Dumas and Stepanov᎐Schmidt criteria as special cases, as shown in Figs. 3Ža. ᎐ Žc.. Case Ža. corresponds to the Eisenstein ᎐Dumas criterion. In this case, Pf has only one point on the dotted vertical line Žwhich means that f has degree n in X and the
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coefficient of X n is a constant in F ., so Pf is contained in a triangle with two vertices on the y-axis. Case Žb. corresponds to the Stepanov᎐Schimdt criterion. In the case Žc., the dotted triangle can be at any position, so Ža. is a special case of it. For an example, the polynomial f s X 2 q y 3 q aX 3 Y q bX 4 y 6 q cX 11 Y 10 has a Newton polytope of type Žc., so it is absolutely irreducible over F for any values of a, b, c. For another example, the polynomial f s X n q Y m q X nq u Y ¨ q X i Y mq j has a Newton polytope of type Žc. whenever gcd Ž n, m . s 1,
j i
-
¨
u
,
where i, u G 1 and j, ¨ G 0. So f is absolutely irreducible. We should emphasize that Theorems 4.2 and 4.11 can be used to build many types of indecomposable polytopes in higher dimensions iteratively from lower dimensions. Thus one can obtain many types of absolutely irreducible polynomials over any given field. Finally, we briefly discuss the relationship of our decomposability to that w13, Chap. 15x. Let P, Q be polytopes in ⺢ n Žnot necessarily of Grunbaun ¨ . integral . Q is said to be homothetic to P if there is a real number t ) 0 and a vector a g ⺢ n such that Q s tP q a s tb q a : b g P 4 . A polytope Q is called homothetically indecomposable if Q s P1 q P2 for any polytopes P1 and P2 then either P1 or P2 is homothetic to Q. Otherwise, Q is called homothetically decomposable. Indecomposable polytopes in this sense have been extensively studied in the literature w8, 16, 26, 27, 40᎐42x. Homothetic decomposability is not comparable with integral decomposability. On the one hand, a triangle is always homothetically indecomposable, but an integral triangle is integrally indecomposable iff the condition in Corollary 4.5 is satisfied. On the other hand, any polygon with more than three edges in the plane is homothetically decomposable, while Fig. 3 shows many integrally indecomposable polygons!
ACKNOWLEDGMENTS The author thanks Joel Brawley, Erich Kaltofen, Jenny Key, Hendrik Lenstra, Jr., and Daqing Wan for their useful comments on an earlier version of the paper and for bringing some of the references to his attention. Thanks is also due to Joe Mott for sending a copy of his paper to the author. A preliminary version of the paper was presented at an Oberwolfach Workshop on Designs and Codes, March 15᎐21, 1998, Germany, and at a Dagstuhl-Seminar on Algorithms and Number Theory, October 26᎐30, 1998, Germany.
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