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Asymptotic Number of Triangulations with Vertices inZ2
Journal of Combinatorial Theory, Series A 86, 200203 (1999) Article ID jcta.1998.2921, available online at http:www.idealibrary.com on
NOTE Asymptotic Number of Triangulations with Vertices in Z 2 S. Yu. Orevkov* Steklov Mathematical Institute, Russian Academy of Science, Gubkina 8, Moscow, 117966, Russia Communicated by the Managing Editors Received December 23, 1997
Let T 2n be the set of all triangulations of the square [0, n] 2 with all the vertices belonging to Z 2. We show that Cn 2
Triangulations with integral vertices appear in the algebraic geometry. They are used in Viro's method of construction of real algebraic varieties with controlled topological properties [2]. In [1], the discriminant of a polynomial a # A c a x a with a fixed finite set of multi-indices A/Z d is described in terms of triangulations of the convex hull of A with vertices in A. Here we study the asymptotics of the number of triangulations with integral vertices when the size of the triangulated polytope tends to infinity. Denote by I n the segment [0, n]/R and let I dn :=I n _ } } } _I n /R d be the d-dimensional cube with the side n. Denote by T dn the set of all triangulations of I dn whose vertices are integral points. Question.
What are the asymptotics of log Card T dn when n Ä ?
Only the evident estimates are known for an arbitrary d 2: A d n d
(A d , B d >0).
To get the left inequality, divide I dn into n d cubes; each of them can be subdivided into simplices at least in two ways. To obtain the right inequality, note that the number of all integral d-simplices contained in I dn is bounded * Partially supported by Grants RFFI-96-01-01218 and DGICYT SAB95-0502.
200 0097-316599 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
NOTE
201
by c 1 n c2 and the number of d-simplices in each T # T dn is bounded by c 3 n d, hence, c3 n d d n
Card T : q=1
c 1 n c2 d c 4(c 1 n c2 ) c3 n q
\ +
(here c 1 , ..., c 4 depend on d but do not depend on n). In this note we show that for d=2 the main term of the asymptotics is const } n 2. A triangulation T # T dn is called primitive if the volume of each d-dimensional simplex equals 1d!. Let PT dn =[T # T dn | T is primitive]. Theorem. There exist positive constants A and B such that An 2
There exist positive constants C and D such that Cn 2
Denote by ?: I 2n Ä I n the projection (x, y) [ x. Given a T # T 2n , let us say that a subset U/I 2n is T-univalent if U is a closed simplicial subcomplex of T such that I n _[0]/U, and all the fibers (? | U ) &1 (x), x # I n are connected. Lemma. Let T # PT 2n and let U be a T-univalent subset of I 2n . If U{I 2n then there exists a triangle _ # T such that _ /3 U and U _ _ is T-univalent. Proof. Denote by T U the set of all triangles _ # T such that _ /3 U and _ & U contains a segment. Let _ # T U . Denote the vertices of _ by a, b, c so that ?(a) ?(b), [ab]/U. We shall say that _ hangs to the left (resp. to the right) if ?(c)
202
NOTE
To each T # PT 2n we associate the sequence of T-univalent subsets I n _[0]=U 0 /U 1 / } } } /U N =I 2n , N=2n 2, where U j+1 =U j _ _ j and _ j is the most left among the triangles _ # T such that _/3 U j , _ _ U j is T-univalent. Denote by E the set of all edges of T and put E j =[e # E | e/U j ], k j =Card E j . Let e 1 , ..., e kN be all the edges of T numerated so that E 0 =[e 1 , ..., e k0 ] and E j+1 "E j =[e kj +1 , ..., e kj+1 ] ( j=0, ..., N), the elements of each E j+1 "E j being numerated from the left to the right. Let us define the vector v T =(v 1 , ..., v kN ) as follows. If e j is vertical or e j / I n _[1] then we put v j =1. Otherwise, if e j =[ab], ?(a)
{
if if if if
?(c)?(b). 2
It is clear that that the number of all possible vectors v T is bounded by 4 3n . Therefore, to complete the proof it suffices to show that T is uniquely determined by v T . Indeed, if v T is known then one can inductively reconstruct all the sets E 0 , E 1 , ... as follows. Suppose U j is already reconstructed. Let e j1 , ..., e jm be all the non-vertical edges (numerated from the left to the right) lying on U j " I 2n . Then the triangle _, adjacent to e ji from above, being attached to U j yields a T-univalent set if and only if one of the following three cases holds: (i) 2 v ji 3; (ii) v ji&1 =4 and v ji =1; (iii) v ji =4 and v ji+1 =1. In all the cases _ is uniquely determined by _ & U j (due to the primitivity condition). K Proof of Corollary. Each T # T 2n can be subdivided to a T $ # PT 2n . Let E and E$ be the sets of edges of T and T $ not lying on I 2n . Clearly that T is uniquely determined by T$ and E. Hence, Card T 2n T $ Card [E/E$] 2 <(Card PT 2n ) } 2 3n since Card E$=3n 2 &2n<3n 2. K The properties of integral points were essential to our proof. However, the following generalization seems to be true. Given a finite set A/R 2, denote by T(A) the set of triangulations of the convex hull of A with vertices belonging to A. Conjecture. There exists a constant C 1 such that logCard T(A) C 1Card A for any finite A/R 2. This is well-known when A is the set of vertices of a convex polygon. In 1 this case Card T(A) is the Catalan number n&1 ( 2n&4 n&2 ) where n=Card A. Note also, that an analogue of the Lemma is true for any finite A/R 2.
NOTE
203
ACKNOWLEDGMENTS I am grateful to V. Kharlamov for suggesting the problem and useful discussions. I thank the referee for useful remarks.
REFERENCES 1. I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinskii, ``Discriminants, Resultants and Multidimensional Determinants,'' Birkhauser, Boston, 1994. 2. O. Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Lecture Notes in Math. 1960 (1984), 187200.
Printed in Belgium
NOTE Asymptotic Number of Triangulations with Vertices in Z 2 S. Yu. Orevkov* Steklov Mathematical Institute, Russian Academy of Science, Gubkina 8, Moscow, 117966, Russia Communicated by the Managing Editors Received December 23, 1997
Let T 2n be the set of all triangulations of the square [0, n] 2 with all the vertices belonging to Z 2. We show that Cn 2
Triangulations with integral vertices appear in the algebraic geometry. They are used in Viro's method of construction of real algebraic varieties with controlled topological properties [2]. In [1], the discriminant of a polynomial a # A c a x a with a fixed finite set of multi-indices A/Z d is described in terms of triangulations of the convex hull of A with vertices in A. Here we study the asymptotics of the number of triangulations with integral vertices when the size of the triangulated polytope tends to infinity. Denote by I n the segment [0, n]/R and let I dn :=I n _ } } } _I n /R d be the d-dimensional cube with the side n. Denote by T dn the set of all triangulations of I dn whose vertices are integral points. Question.
What are the asymptotics of log Card T dn when n Ä ?
Only the evident estimates are known for an arbitrary d 2: A d n d
(A d , B d >0).
To get the left inequality, divide I dn into n d cubes; each of them can be subdivided into simplices at least in two ways. To obtain the right inequality, note that the number of all integral d-simplices contained in I dn is bounded * Partially supported by Grants RFFI-96-01-01218 and DGICYT SAB95-0502.
200 0097-316599 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.
NOTE
201
by c 1 n c2 and the number of d-simplices in each T # T dn is bounded by c 3 n d, hence, c3 n d d n
Card T : q=1
c 1 n c2 d c 4(c 1 n c2 ) c3 n q
\ +
(here c 1 , ..., c 4 depend on d but do not depend on n). In this note we show that for d=2 the main term of the asymptotics is const } n 2. A triangulation T # T dn is called primitive if the volume of each d-dimensional simplex equals 1d!. Let PT dn =[T # T dn | T is primitive]. Theorem. There exist positive constants A and B such that An 2
There exist positive constants C and D such that Cn 2
Denote by ?: I 2n Ä I n the projection (x, y) [ x. Given a T # T 2n , let us say that a subset U/I 2n is T-univalent if U is a closed simplicial subcomplex of T such that I n _[0]/U, and all the fibers (? | U ) &1 (x), x # I n are connected. Lemma. Let T # PT 2n and let U be a T-univalent subset of I 2n . If U{I 2n then there exists a triangle _ # T such that _ /3 U and U _ _ is T-univalent. Proof. Denote by T U the set of all triangles _ # T such that _ /3 U and _ & U contains a segment. Let _ # T U . Denote the vertices of _ by a, b, c so that ?(a) ?(b), [ab]/U. We shall say that _ hangs to the left (resp. to the right) if ?(c)
202
NOTE
To each T # PT 2n we associate the sequence of T-univalent subsets I n _[0]=U 0 /U 1 / } } } /U N =I 2n , N=2n 2, where U j+1 =U j _ _ j and _ j is the most left among the triangles _ # T such that _/3 U j , _ _ U j is T-univalent. Denote by E the set of all edges of T and put E j =[e # E | e/U j ], k j =Card E j . Let e 1 , ..., e kN be all the edges of T numerated so that E 0 =[e 1 , ..., e k0 ] and E j+1 "E j =[e kj +1 , ..., e kj+1 ] ( j=0, ..., N), the elements of each E j+1 "E j being numerated from the left to the right. Let us define the vector v T =(v 1 , ..., v kN ) as follows. If e j is vertical or e j / I n _[1] then we put v j =1. Otherwise, if e j =[ab], ?(a)
{
if if if if
?(c)?(b). 2
It is clear that that the number of all possible vectors v T is bounded by 4 3n . Therefore, to complete the proof it suffices to show that T is uniquely determined by v T . Indeed, if v T is known then one can inductively reconstruct all the sets E 0 , E 1 , ... as follows. Suppose U j is already reconstructed. Let e j1 , ..., e jm be all the non-vertical edges (numerated from the left to the right) lying on U j " I 2n . Then the triangle _, adjacent to e ji from above, being attached to U j yields a T-univalent set if and only if one of the following three cases holds: (i) 2 v ji 3; (ii) v ji&1 =4 and v ji =1; (iii) v ji =4 and v ji+1 =1. In all the cases _ is uniquely determined by _ & U j (due to the primitivity condition). K Proof of Corollary. Each T # T 2n can be subdivided to a T $ # PT 2n . Let E and E$ be the sets of edges of T and T $ not lying on I 2n . Clearly that T is uniquely determined by T$ and E. Hence, Card T 2n T $ Card [E/E$] 2 <(Card PT 2n ) } 2 3n since Card E$=3n 2 &2n<3n 2. K The properties of integral points were essential to our proof. However, the following generalization seems to be true. Given a finite set A/R 2, denote by T(A) the set of triangulations of the convex hull of A with vertices belonging to A. Conjecture. There exists a constant C 1 such that logCard T(A) C 1Card A for any finite A/R 2. This is well-known when A is the set of vertices of a convex polygon. In 1 this case Card T(A) is the Catalan number n&1 ( 2n&4 n&2 ) where n=Card A. Note also, that an analogue of the Lemma is true for any finite A/R 2.
NOTE
203
ACKNOWLEDGMENTS I am grateful to V. Kharlamov for suggesting the problem and useful discussions. I thank the referee for useful remarks.
REFERENCES 1. I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinskii, ``Discriminants, Resultants and Multidimensional Determinants,'' Birkhauser, Boston, 1994. 2. O. Ya. Viro, Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Lecture Notes in Math. 1960 (1984), 187200.
Printed in Belgium