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Roots of Affine Polynomials
Annals of Discrete Mathematics 30 (1986) 303-310
0 Elsevier Science Puhlishers B.V (North-Holland)
303
ROOTS OF AFFINE POLYNOMIALS
Giampaolo M e n i c h e t t i (-) D i p a r t i m e n t o d i Matematica, U n i v e r s i t P d i Bologna, I t a l y
INTRODUCTION.
Let F = GF(q) be a G a l o i s f i e l d of o r d e r q = p h , where p i s a p r i m e ,
and l e t K = GF(qn) be an a l g e b r a i c e x t e n s i o n of a g i v e n d e g r e e n > l . An a f f i n e
pziynon;iaZ (of K[x]over F ) i s a polynomial of type (1)
P ( x ) = L(x) - b , b € K ,
with
...,un-1 }
I f a b a s i s {uo,ul,
n- 1 of K o v e r F i s f i x e d then we can p u t x = 5 x . u . .
i =O
1 1
Hence, t h e d e t e r m i n a t i o n of ( e v e n t u a l ) r o o t s of t h e polynomial ( 1 ) i n K can be reduced t o t h c d e t e r m i n a t i o n of s o l u t i o n s o f a l i n e a r s y s t e m of e q u a t i o n s i n i n d e t e r m i n a t e s xi and w i t h c o e f f i c i e n t s i n F ( c f . [ l l , Chap.11). This procedure i s , however, t e d i o u s a l s o i n t h e most simply c a s e s and does n o t d e c i s e "a p r i o r i " how many r o o t s i n K e x i s t . I n t h i s p a p e r , we prove t h a t the e q u a t i o n ( 1 ) h a s r o o t s i n K i f and o n l y i f the following system of l i n e a r equations
(3)
+...+
llYl
loyo
+
l:-lYO
+ l:yl
= b
+. . .+ 1 ~ - 2 y n - l = bq
.....................
1qn-ly0 + 1;
1 n-1Yn-1
n- 1
y1 +.
.. +
I
.
1;
,
.
n-1
n- 1
yn-l = bq
has s o l u t i o n s . Moreover, i f ( 3 ) i s s o l v a b l e and i f r is t h e rank of t h e m a t r i c e s b e l o n g i n g t o ( 3 ) , then qn-'
g i v e s us t h e number of s o l u t i o n s of ( 1 ) i n K. Besides
t h i s , we show t h a t t h e r o o t s of ( 1 ) a r e e x p r e s s i b l e as f u n c t i o n s of c e r t a i n s o l u t i o n s of t h e l i n e a r s y s t e m ( 3 ) . I n p a r t i c u l a r , t h e o b t a i n e d r e s u l t s a r e u s e f u l a l s o i n t h e c a s e t h a t t h e c o e f f i c i e n t s of t h e polynomial ( 1 ) are n o t c o n s t a n t ( c f . f . e . [2] ).
(-) This r e s e a r c h was s u p p o r t e d i n p a r t by a g r a n t from t h e M.P.I.(40% f u n d s ) .
304
G.Menicketti 1. An n x n matrix of the type
.
A( .a ,al ,. . ,an-
... an-1
)=
is called autocirculant. Each row of A(ao,al,...,an-l) is obtained by the previous one
if we permute the elements under the cyclic permutation
apply the field automorphism a : K
-+
(0
1. ..n-1) and then
K , a I+ aq.
The sum and the product of autocirculant matrices are autocirculant matrices. In addition, if
A
is autocirculant, the transpose
it is
also autocirculant.
Let
T = and Aq(aO,al
...,0 )
&(O,l,O,
,...,an-1)
=
A(aq,aq, ...,a:-1). 0 1
It is easy to verify
T A- T-l , Aq= -
(4)
LEMMA 1. An autocirculant matrix
A has
rank r, 1 5 r G n - 1 , i f and only if i t s f i r s t r FOWS (colwnns) are l i n e a r l y K-independent and i t s (r+l)-th row (column) is a K-linear cornbination o f t h e preceding POWS (coLwmzs). Proof. The transpose of an autocirculant matrix is itself autocirculant. Thus it is sufficient to prove the statement for the columns of A.
The assertion is an obvious consequence of the following observation, Let
A=
(%,,A1,...,$I-l).
~f
s-1 (5) --s A = E kiAi , kfK, i=O s-1 0 < s < n-1, then A' = C kqAq and therefore -s
i=o
1-1
s-1 s-1 1 k . A . = c k!A. ,k!E K. 1 kqii+l + kq s-liz0 l-li=o 1-1 1 i=O 2 3 n-s-1 and use previous If we raise both sides of ( 5 ) t o the powers q ,q ,...,q
s-1
s-2
&+I=. 1=0 1 kqAi+l
=
arguments, we see that the columns &+2, K-linear combination of
%, A1,...,s A .n
& + 3 , , , , , &-l
can be expressed as a
PROPOSITION 2 . I$& i s an autocirculant m a t r i x o f rank r then the homogeneous
Linear system (6)
-AY
=
0 ,y =
(Yo Yl
Y,-l)t
..,
t zl,,. z 0 0) w i t h zr # 0 . Furthermore, n-r and LL' t = 2. given A' = A(zo, zl,. . , z r' 0,. , O ) , w i h m e rank(&')= Proof. If one has A = (A+, . , 411) then Lemma 1 guaranties the
has s o l u t i o n s of the type 5
.
= (zo
.. 4,. .
existence of the elements a ! E K such that A
-K
n-1
=
T a!A. holds. And, this proves the
i=o
1-1
Roots of Affine Polynomials
305
f i r s t p a r t of t h e a s s e r t i o n .
(s,A; ,..., A&). n- 1 By d e f i n i t i o n , one h a s % T-lzq, - ..., r-(n-1IzqLet Att=
-A'= 1
- and
therefore
= z
A= -l'
A?= 0
If we c o n s i d e r t h e q-th power of deduce
0=
=
TAAi.
Thus
and t a k e i n account a l s o ( 4 ) , w e
h i = 0.Analogously,
--2
and have f i n a l l y
hi,. . . , g-l)= Ah'
A($,
=
0,. , . ,A$-l=g,
we prove A A ' =
0.
A;,
We deduce, i n p a r t i c u l a r , t h a t e v e r y element Ai,, ..., -nA ' €Kn i s a 1 s o l u t i o n of t h e l i n e a r s y s t e m ( 6 ) and h e n c e , i t f o l l o w s r a n k ( A ' t ) , < n - r . To see t h e i n e q u a l i t y r a n k ( A ' ) > , n - r ,
w e remember t h a t t h e m a t r i x which a g r e e s
i n t h e f i r s t n-r rows and l a s t n-r columns w i t h
..
A'
is non-singular.0
LEMMA 3 . T4e elements w .€ K , i = 0,1,. ,s , are l i n e a r l y F-independent if und n-1 t n o n l y if the vectoras w . = (wi wq w! ) EK , i = 0,1,. ,s, are Zinearly
.. .
-1
..
K-independent . Proof. Let us examine t h e c o n d i t i o n S
1 k.w.
(7)
i =O
1-1
=
0,kiEK,
under t h e h y p o t h e s i s t h a t t h e e l e m e n t s wi a r e F-independent. I f w e suppose t h a t a t l e a s t one c o e f f i c i e n t k i , then w e can d e t e r m i n e k E K such t h a t h = kk 0
obtain S
T: h.w. 1-1
i=O
= 0 , h . = kk
-
I
0'
f o r example ko, i s n o t z e r o ,
t r ( h o ) # 0 ( ' ) h o l d s . From ( 7 ) ,
i'
s
R a i s i n g t h e l e f t s i d e o f t h i s e q u a t i o n t o t h e powers qJ we o b t a i n ? h: i =O
...,n-1.
j = 0,1,
particular
S
I f we add t h e s e n e x p r e s s i o n s , w e f i n d C ( t r ( h i ) ) w i = i =O
j
wi
2;
=
we
0,
in
S
Y ( t r ( h i ) ) w i = 0, t r ( h o ) # 0 , i=O i n contrast with the hypothesis.
It i s e v i d e n t how t h e second p a r t of t h e t h e s i s may be proved.! COROLLARY 4 . Y7ie n x n matrix
nm-singular
2
=
(%,El,... ,%-,),
if and onlg if { u o , ul,. . .
...
n-1
)t,iz li= ( u i :u uqi , u ~ - ~is} a b a s i s o f the vector F-space K.!
For any polynomial ( 2 ) , t h e s e t Z(L) = { x € K : L(x) =
01
is o b v i o u s l y a v e c t o r subspace of K . Moreover, i f Z(P) = { x E K : P ( x ) = 01
(8)
(l)
Z(P)
=
xo+ Z ( L ) , x0E Z ( P ) .
t r ( x ) = t r ( x ) = x + xq + F
...
+ xq
n-1
,v
XEK.
# d then
G. Menichetti
306
Given an a f f i n c polynomial ( l ) , l e t
A(P)
= G(L):
PROPOSITION 5.
= A(lo,ll,...,ln-l).
If rank(A(L))
...,w
Proof. Let {wo,wl,
r then d i n $ Z ( L )
=
=
n-r.
] b e a b a s i s of Z(L) and l e t V C K n be t h e s o l u t i o n
space of t h e homogeneous l i n e a r s y s t e m
A(L)JI-=
(9)
0,11. =
.. . yn-l)t.
(yo y1
From Lemma 3 , i t f o l l o w s t h a t the v e c t o r s w.= (w. wq -1
1
1
.,. w:
n-1
,...,s ,
)t,i=O,l
a r e l i n e a r l y K-independent and i t i s e a s i l y v e r i f i e d t h a t each of them i s a s o l u t i o n of ( 9 ) . Thus,
Let
(11)
<
di%Z(L)
(10)
A' be
di%V = n-r.
an a u t o c i r c u l a n t m a t r i x which s a t i s f i e s t h e c o n d i t i o n s
-
i \ ( L ) A f t = 0 , r a n k ( A ' ) = n-r
2 = (%,,ul, ...,-n-1 u )
( c f . P r o p . 2 ) and l e t Coroll. 4 ) .
be an n x n non-singular m a t r i x ( c f .
From ( l l ) , we deduce
A(L)(A'
t
g)
=
With t h e o b s e r v a t i o n
0,rank(&' t 2) = n-r. t h a t Attg = (I&,?; ,...,$I&), u! -1 i
can conclude t h a t u ! E Z ( L ) ,
O,l,
=
(lo),
( c f . a l s o Lemma 3 ) . From t h i s and immediately
...,n-1
.o
(u! u!
=
1
and d i % < u ; ) , u i
'... u!'
,...,u'n-1 1
n-1
) t , we
> = n-r
t h e P r o p o s i t i o n 5 f o l l o w now
COROLLARY 6 . Suppot~e rank(A(L)) = r .
Tf
-z =
0 O...O)
zl...z
(zo
t
is a
s d l u i i o n of the Zineura system ( 9 ) f o r any choose of t h e basis I u o , u l , . , , , u
n-l
o f tlie vector F-space K, the eZements (12)
x. 1
= z
n-r
u + O i
29,
us
n-r
n-r+l n-r+l + zq uq + r-1 1
... +
n-1 2;
u4
n-1
,
i=O,l,
1
...,n-1,
f o m n s e t of generators of Z(L). Hence, oiie has (12)'
Z(L) = { x = z k O
n-r
+ zqr
n-r
kq
+
n-1
... + zq1
n-1
kq
: kEK}.
Proof.
The C o r o l l a r y f o l l o w s from t h e proof o f t h e p r e v i o u s P r o p o s i t i o n i f n-r n-r+l n- 1 one o b s e r v e s t h a t A't= A(zo,O 0,z: .z:-~ zq ). 1 t PROPOSITION 7 . If = ( z o z l . . . z ~ - ~ E) K" is a sokction of the linear system
z
(13)
A(L)y =
b, y
,....
,...,
= ( y o y1
...
=
(b bq
... bq
0
n-1
then, for every v E K w i t h t r ( v ) # 0 , (14)
x = (z v 0 0
+
z:-lvq
2
+ zq vq n-2
2
+...+
n-1 n-1 zq vq )/tr(v) 1
It,
Roots of Affine Polynomials
307
is a root of the poZynomiaZ ( 1 ) . P r o o f . Let
A(?)
-At (5)=
=
A(zo,zl,,s.,zn-l).
(2, -'zq, T 2 z-q
Raising L ( L ) z =
b
2
Then
,..., -T - ( n - l ) z q -
n- 1
).
- = Lq o r A(L)(X-'L~)
t o t h e power q , we o b t a i n A q ( L ) z q =bq=(bq bq
Using ( 4 ) , we h a v e , t h e r e f o r e , -TA(L)L1zq
=
2.
2
...bq
n-1
b)t.
Iterating this,
we f i n d
Thus, i t f o l l o w s , t A(L)A ( 2 )
(b b
=
... b).
Now, t h e r i g h t m u l t i p l i c a t i o n o f t h i s e q u a t i o n by n-1 ( t r ( v ) ) b ,I1= (v' v'q., , v f q
A ( L )1'=
=
( v vq...
n-1 vq ) t gives
lt
COROLLARY 8. The poZyizorniaZ (1) hus r o o t s i ? i K ,if and onZy if rank(A(1,)) =
rank(A(L) Ib-)
=
r . If t h i s c ondition holds, one izus IZ(P)I
= qn-r.
Proof. I f ( 1 ) h a s a r o o t x E K t h e n r a i s i n g b o t h s i d e s of t h e e q u a l i t y 0
t o t h e powers q , q
Thus,
&=
n-1
(xo
,..., q n-1 , we
... + 1:-2x: .....................
find
n- 1
= bq,
+ '10 xq0 +
l:-l~o
:1
2
xi +
n- 1
xo+ 1; X:
...
...
+ :1
n-1
n-1 n-1 xq = bq 0
n- 1 xq ) t is a s o l u t i o n of ( 1 3 ) . From h e r e and from Prop.7, 0
it
f o l l o w t h a t (1) h a s r o o t s i n K i f and o n l y i f ( 1 3 ) h a s s o l u t i o n s .
Taking i n account (8), t h e l a s t p a r t o f t h e a s s e r t i o n f o l l o w s from Prop.5.u In p a r t i c u l a r , we find the following RESULT (Dickson 1 3 ) ) .
c r n d onz$
If d e t ( A -( L ) ) #
T k ma[) L : K
-t
K, x
+
L ( x ) is n p r m u t a t i o n on K f,f
0.
Moreover, we o b s e r v e t h a t i f d e t ( A ( L ) ) # 0 , t h e o n l y r o o t x E K of t h e 0
polynomial ( 1 ) can be determined u s i n g Cramer's r u l e , t h a t i s
x 0= d e t ( b, ($,
Al.... ,&-l)
4 ,...,$-l)/det(%, =
A19 .
-
*
sS-1)
3
A(L).
I n g e n e r a l , t h e a f f i n e s u b v a r i e t y of R c o n s i s t i n g o f t h e s o l u t i o n s of polynomial ( 1 ) is given by (8) w i t h xo and Z ( L ) e x p r e s s e d by ( 1 4 ) and ( 1 2 ) ' respectively
.
308
G.Meniclietti From C o r o l l a r y 8 , we deduce t h e f o l l o w i n g u s e f u l
OBSERVATION. A polynomiaz (1) W i t h d e g ( L ( x ) )
=
q d , 0 ,< d
corripZetcZy redueible in K if and on2y if r a n k ( L ( 1 ) j b )
=
<
n-1, is
rank(A(L)) = n-d.
Another consequence i s t h e f o l l o w i n g
PROPOSITION 9 . Tuo a f f i n e poZynomiaZs, (1) and P ' ( x )
=
L'(x)
-
b ' , have
common u>oots in K if and onZy ?'f t h e equalions of t h e Zinear sistems (13) and A(L') -
y= b ' m e compatible. Proof. If x E K i s acommon r o o t of b o t h P(x) and P'(x) t h e n x
(xo :x
n-1
... x:
4
O
=
) t i s a s o l u t i o n f o r both l i n e a r systems i n t h e a s s e r t i o n .
Conversely, i f t h e e q u a t i o n s of b o t h systems a r e c o m p a t i b l e , we f i n d , by ( 1 4 ) , a common r o o t f o r t h e given polynomials.[ I t i s easy t o prove t h a t , when t h e c o n d i t i o n of t h e p r e v i o u s p r o p o s i t i o n i s
s a t i s f i e d , t h e s e t of common r o o t s f o r P(x) and P ' ( x ) i s an a f f i n e s u b v a r i e t y of
,-;I
K whose dimension i s n - r ' ,
(-Gi1:
A(L)
r ' = rank
-
where =
(-:
A(L)
)
b
-I - :. A(L')I b'
rank
--
Now we want t o use t h e p r e v i o u s r e s u l t s t o d i s c u s s t h e e q u a t i o n
xq
(15)
m
- x
=
b , b E K , 1 ,< m , < n-1
.
F i r s t we observe t h a t , given d = (n,m) and k = n / d , t h e i n t e g e r s i m + j ,
i
= O,l,
...,k-1,
j = O , l , . . .,d-1,
a r e p a i r w i s e incongruent modulo n.
I n t h i s c a s e , t h e l i n e a r s y s t e m (13) becomes
(16)
Y2m+ j
= bq'
- 'm+j
...................
'( k-1 ) m+ j
-
(k-2 )m+j
'i
-
(k-1 )m+ j
'+m
=
'+(k-2)m bqJ
=
bq
j + (k-1 ) m
,
j = 0.1,
and t h u s i t s e q u a t i o n s a r e compatible i f and o n l y i f
...,d-1, im j k-1 j+im k-1 C bq = ( C bq )' = 0. i=O
i=O
From t h i s , we deduce t h a t (15) has some r o o t s i n K i f and o n l y i f (17)
k-1 im C bq = t r F , ( b ) = 0,
i =O
d where F' = GF(q )
(')
C_
GF(qn)
(2),
The i n t e g e r s h d , h = 0 , 1 ,
modulo m and t h e r e f o r e
k-1
I: bq
i =O
... ,k-1, im
k-1 =
and i m , i = 0,1,
C bq h=O
hd
.
...,k-1,
a r e congruent
Roots of Affine Polynomials m
-
L(x) = xq
309
x implies obviously
d Z ( L ) = GF(q ) , d = ( n , m ) .
(18)
T h e r e f o r e , w e can d e t e r m i n e a r o o t x E K of (15) u s i n g P r o p . 7 and supposing t h a t C
(17) i s s a t i s f i e d .
From (16), by s u c c e s s i v e s u b s t i t u t i o n s , we f i n d yim+l
i-1
= yj + (
C bq
h=O
and by (17) yim+j =
A. 1
hm
k-1
C bq
(
hm
h=i
)q
)'
j
j
, i
1,2
=
, X.EK,
,...,k-1,
i = 0,1,
J
...,d-1,
j = 0,1,
...,k-1,
j = O,l,...,d-l
Let u s c o n s i d e r t h e p a r t i c u l a r s o l u t i o n =
'im+j
X
obtained f o r
j
-
(
k-1
C bq
hm
h=i
)q
j
, i
O,l,
=
...,k-1,
j
=
...,d-1.
= 0, j = 0,1,
From ( 1 4 ) w e o b t a i n
x tr(v) 0
n-1 =
h
E 24 vq n-h
h=O d- 1
Hence, s e t t i n g v = wq
, we
h
'
k-ld-1 =
i=O
qn-(im+j)
j=o 'im+j
n-(im+j)
"4
have
k-1 d-1 n-(im+j) n-(im+j)+d-1 C C z;m+j wq i=O j=O where t r ( w ) = t r ( v ) # 0.
x tr(w) 0
=
I f we o b s e r v e t h a t n-(im+j) = -
zSm+j
rm k-1 n+(h-i)m k-i-1 Cbq = C bq , h=i r=O
t h e n , s u b s t i t u t i n g i n t o t h e p r e v i o u s e q u a l i t y , one h a s
x tr(w) 0
rm d-1 n-im+(d-1-j) k-1 k-i-1 Z C bq C wq i = O r=O j =O k-1 k-i-1 r m d-1 n-im+s = - C C bq C w q i = O r=O s =o k-1 k-i-1 rm d-1 s n-im = - C Z bq ( C w ' ) ~ i=O r=O s=o =
-
.
From h e r e , p u t t i n g a =
d-1 C
s W
~
s=o
and o b s e r v i n g
tr(w) =
we deduce
k-1 d-1
c
i=o
I: wq
j=O
im+j
k-1 =
c
i=O
a'
im
= trF,(a),
...,d-1,
0,1,
.
G.Menichetti
310 k-1 C i=O k = C h=1 k-1
xOtrFI(a) =
-
= -
c
k-i-1 rm n-im C bq aq r=O r m hm h-1 C b q aq
r=O
h-1 ~
h = l r=O
b
rm
q aq
hm
.
T h e r e f o r e : The equation (15) has r o o t s i n K = GF(qn) i f and only i f b s a t i s f i e s
the condition ( 1 7 ) . If such condition i s s a t i s f i e d , the s e t of r o o t s i s the a f f i n e subvariety ( 8 ) in which Z ( L ) i s given by (18) and x
=--
k-1 h-1 rm hm C C b q a' trF,(a) h=l r=O
,
t r F l ( a ) # 0.
I f ( k , p ) = 1 ( p = c h a r K ) then t r F l ( l ) = k # 0 and t h e r e f o r e , we can s e t a = l . The p r e v i o u s r e s u l t a l l o w s u s t o determine t h e r o o t s of a second d e g r e e e q u a t i o n i n a f i e l d K of c h a r 2 . I n f a c t , f o r q = 2 , m = 1, w e f i n d t h e w e l l known c o n d i t i o n t r ( b ) = 0 i n o r d e r t h a t t h e e q u a t i o n X'
+ x
t
b = 0
h a s a r o o t in K = GF(2").
Moreover, from (18) and ( 1 9 ) , we deduce t h a t t h e r o o t s
of t h e above e q u a t i o n a r e x
n-1 =--
C
h-1
C b
t r ( a ) h=l r=O
2r 2h a
and
xo+ 1 ,
where a E K i s a f i x e d element w i t h t r ( a ) # 0.
REFERENCES
[ l ] Berlekamp, E . R . ,
AZgebraic coding theory (Mc Graw Book Company,New York,1968).
121 B i l i o t t i M. and M e n i c h e t t i G . , On a g e n e r a l i z a t i o n of Kantor's l i k e a b l e planes, Geom. D e d i c a t a , 1 7 (1985) 253-277.
[ 3 ] Dickson, L . E . ,
Linear Groups w i t h an e x p o s i t i o n o f t h e Galois fieZd theory
(Teubner, L e i p z i g . R e p r i n t Dover, New York, 1958).
0 Elsevier Science Puhlishers B.V (North-Holland)
303
ROOTS OF AFFINE POLYNOMIALS
Giampaolo M e n i c h e t t i (-) D i p a r t i m e n t o d i Matematica, U n i v e r s i t P d i Bologna, I t a l y
INTRODUCTION.
Let F = GF(q) be a G a l o i s f i e l d of o r d e r q = p h , where p i s a p r i m e ,
and l e t K = GF(qn) be an a l g e b r a i c e x t e n s i o n of a g i v e n d e g r e e n > l . An a f f i n e
pziynon;iaZ (of K[x]over F ) i s a polynomial of type (1)
P ( x ) = L(x) - b , b € K ,
with
...,un-1 }
I f a b a s i s {uo,ul,
n- 1 of K o v e r F i s f i x e d then we can p u t x = 5 x . u . .
i =O
1 1
Hence, t h e d e t e r m i n a t i o n of ( e v e n t u a l ) r o o t s of t h e polynomial ( 1 ) i n K can be reduced t o t h c d e t e r m i n a t i o n of s o l u t i o n s o f a l i n e a r s y s t e m of e q u a t i o n s i n i n d e t e r m i n a t e s xi and w i t h c o e f f i c i e n t s i n F ( c f . [ l l , Chap.11). This procedure i s , however, t e d i o u s a l s o i n t h e most simply c a s e s and does n o t d e c i s e "a p r i o r i " how many r o o t s i n K e x i s t . I n t h i s p a p e r , we prove t h a t the e q u a t i o n ( 1 ) h a s r o o t s i n K i f and o n l y i f the following system of l i n e a r equations
(3)
+...+
llYl
loyo
+
l:-lYO
+ l:yl
= b
+. . .+ 1 ~ - 2 y n - l = bq
.....................
1qn-ly0 + 1;
1 n-1Yn-1
n- 1
y1 +.
.. +
I
.
1;
,
.
n-1
n- 1
yn-l = bq
has s o l u t i o n s . Moreover, i f ( 3 ) i s s o l v a b l e and i f r is t h e rank of t h e m a t r i c e s b e l o n g i n g t o ( 3 ) , then qn-'
g i v e s us t h e number of s o l u t i o n s of ( 1 ) i n K. Besides
t h i s , we show t h a t t h e r o o t s of ( 1 ) a r e e x p r e s s i b l e as f u n c t i o n s of c e r t a i n s o l u t i o n s of t h e l i n e a r s y s t e m ( 3 ) . I n p a r t i c u l a r , t h e o b t a i n e d r e s u l t s a r e u s e f u l a l s o i n t h e c a s e t h a t t h e c o e f f i c i e n t s of t h e polynomial ( 1 ) are n o t c o n s t a n t ( c f . f . e . [2] ).
(-) This r e s e a r c h was s u p p o r t e d i n p a r t by a g r a n t from t h e M.P.I.(40% f u n d s ) .
304
G.Menicketti 1. An n x n matrix of the type
.
A( .a ,al ,. . ,an-
... an-1
)=
is called autocirculant. Each row of A(ao,al,...,an-l) is obtained by the previous one
if we permute the elements under the cyclic permutation
apply the field automorphism a : K
-+
(0
1. ..n-1) and then
K , a I+ aq.
The sum and the product of autocirculant matrices are autocirculant matrices. In addition, if
A
is autocirculant, the transpose
it is
also autocirculant.
Let
T = and Aq(aO,al
...,0 )
&(O,l,O,
,...,an-1)
=
A(aq,aq, ...,a:-1). 0 1
It is easy to verify
T A- T-l , Aq= -
(4)
LEMMA 1. An autocirculant matrix
A has
rank r, 1 5 r G n - 1 , i f and only if i t s f i r s t r FOWS (colwnns) are l i n e a r l y K-independent and i t s (r+l)-th row (column) is a K-linear cornbination o f t h e preceding POWS (coLwmzs). Proof. The transpose of an autocirculant matrix is itself autocirculant. Thus it is sufficient to prove the statement for the columns of A.
The assertion is an obvious consequence of the following observation, Let
A=
(%,,A1,...,$I-l).
~f
s-1 (5) --s A = E kiAi , kfK, i=O s-1 0 < s < n-1, then A' = C kqAq and therefore -s
i=o
1-1
s-1 s-1 1 k . A . = c k!A. ,k!E K. 1 kqii+l + kq s-liz0 l-li=o 1-1 1 i=O 2 3 n-s-1 and use previous If we raise both sides of ( 5 ) t o the powers q ,q ,...,q
s-1
s-2
&+I=. 1=0 1 kqAi+l
=
arguments, we see that the columns &+2, K-linear combination of
%, A1,...,s A .n
& + 3 , , , , , &-l
can be expressed as a
PROPOSITION 2 . I$& i s an autocirculant m a t r i x o f rank r then the homogeneous
Linear system (6)
-AY
=
0 ,y =
(Yo Yl
Y,-l)t
..,
t zl,,. z 0 0) w i t h zr # 0 . Furthermore, n-r and LL' t = 2. given A' = A(zo, zl,. . , z r' 0,. , O ) , w i h m e rank(&')= Proof. If one has A = (A+, . , 411) then Lemma 1 guaranties the
has s o l u t i o n s of the type 5
.
= (zo
.. 4,. .
existence of the elements a ! E K such that A
-K
n-1
=
T a!A. holds. And, this proves the
i=o
1-1
Roots of Affine Polynomials
305
f i r s t p a r t of t h e a s s e r t i o n .
(s,A; ,..., A&). n- 1 By d e f i n i t i o n , one h a s % T-lzq, - ..., r-(n-1IzqLet Att=
-A'= 1
- and
therefore
= z
A= -l'
A?= 0
If we c o n s i d e r t h e q-th power of deduce
0=
=
TAAi.
Thus
and t a k e i n account a l s o ( 4 ) , w e
h i = 0.Analogously,
--2
and have f i n a l l y
hi,. . . , g-l)= Ah'
A($,
=
0,. , . ,A$-l=g,
we prove A A ' =
0.
A;,
We deduce, i n p a r t i c u l a r , t h a t e v e r y element Ai,, ..., -nA ' €Kn i s a 1 s o l u t i o n of t h e l i n e a r s y s t e m ( 6 ) and h e n c e , i t f o l l o w s r a n k ( A ' t ) , < n - r . To see t h e i n e q u a l i t y r a n k ( A ' ) > , n - r ,
w e remember t h a t t h e m a t r i x which a g r e e s
i n t h e f i r s t n-r rows and l a s t n-r columns w i t h
..
A'
is non-singular.0
LEMMA 3 . T4e elements w .€ K , i = 0,1,. ,s , are l i n e a r l y F-independent if und n-1 t n o n l y if the vectoras w . = (wi wq w! ) EK , i = 0,1,. ,s, are Zinearly
.. .
-1
..
K-independent . Proof. Let us examine t h e c o n d i t i o n S
1 k.w.
(7)
i =O
1-1
=
0,kiEK,
under t h e h y p o t h e s i s t h a t t h e e l e m e n t s wi a r e F-independent. I f w e suppose t h a t a t l e a s t one c o e f f i c i e n t k i , then w e can d e t e r m i n e k E K such t h a t h = kk 0
obtain S
T: h.w. 1-1
i=O
= 0 , h . = kk
-
I
0'
f o r example ko, i s n o t z e r o ,
t r ( h o ) # 0 ( ' ) h o l d s . From ( 7 ) ,
i'
s
R a i s i n g t h e l e f t s i d e o f t h i s e q u a t i o n t o t h e powers qJ we o b t a i n ? h: i =O
...,n-1.
j = 0,1,
particular
S
I f we add t h e s e n e x p r e s s i o n s , w e f i n d C ( t r ( h i ) ) w i = i =O
j
wi
2;
=
we
0,
in
S
Y ( t r ( h i ) ) w i = 0, t r ( h o ) # 0 , i=O i n contrast with the hypothesis.
It i s e v i d e n t how t h e second p a r t of t h e t h e s i s may be proved.! COROLLARY 4 . Y7ie n x n matrix
nm-singular
2
=
(%,El,... ,%-,),
if and onlg if { u o , ul,. . .
...
n-1
)t,iz li= ( u i :u uqi , u ~ - ~is} a b a s i s o f the vector F-space K.!
For any polynomial ( 2 ) , t h e s e t Z(L) = { x € K : L(x) =
01
is o b v i o u s l y a v e c t o r subspace of K . Moreover, i f Z(P) = { x E K : P ( x ) = 01
(8)
(l)
Z(P)
=
xo+ Z ( L ) , x0E Z ( P ) .
t r ( x ) = t r ( x ) = x + xq + F
...
+ xq
n-1
,v
XEK.
# d then
G. Menichetti
306
Given an a f f i n c polynomial ( l ) , l e t
A(P)
= G(L):
PROPOSITION 5.
= A(lo,ll,...,ln-l).
If rank(A(L))
...,w
Proof. Let {wo,wl,
r then d i n $ Z ( L )
=
=
n-r.
] b e a b a s i s of Z(L) and l e t V C K n be t h e s o l u t i o n
space of t h e homogeneous l i n e a r s y s t e m
A(L)JI-=
(9)
0,11. =
.. . yn-l)t.
(yo y1
From Lemma 3 , i t f o l l o w s t h a t the v e c t o r s w.= (w. wq -1
1
1
.,. w:
n-1
,...,s ,
)t,i=O,l
a r e l i n e a r l y K-independent and i t i s e a s i l y v e r i f i e d t h a t each of them i s a s o l u t i o n of ( 9 ) . Thus,
Let
(11)
<
di%Z(L)
(10)
A' be
di%V = n-r.
an a u t o c i r c u l a n t m a t r i x which s a t i s f i e s t h e c o n d i t i o n s
-
i \ ( L ) A f t = 0 , r a n k ( A ' ) = n-r
2 = (%,,ul, ...,-n-1 u )
( c f . P r o p . 2 ) and l e t Coroll. 4 ) .
be an n x n non-singular m a t r i x ( c f .
From ( l l ) , we deduce
A(L)(A'
t
g)
=
With t h e o b s e r v a t i o n
0,rank(&' t 2) = n-r. t h a t Attg = (I&,?; ,...,$I&), u! -1 i
can conclude t h a t u ! E Z ( L ) ,
O,l,
=
(lo),
( c f . a l s o Lemma 3 ) . From t h i s and immediately
...,n-1
.o
(u! u!
=
1
and d i % < u ; ) , u i
'... u!'
,...,u'n-1 1
n-1
) t , we
> = n-r
t h e P r o p o s i t i o n 5 f o l l o w now
COROLLARY 6 . Suppot~e rank(A(L)) = r .
Tf
-z =
0 O...O)
zl...z
(zo
t
is a
s d l u i i o n of the Zineura system ( 9 ) f o r any choose of t h e basis I u o , u l , . , , , u
n-l
o f tlie vector F-space K, the eZements (12)
x. 1
= z
n-r
u + O i
29,
us
n-r
n-r+l n-r+l + zq uq + r-1 1
... +
n-1 2;
u4
n-1
,
i=O,l,
1
...,n-1,
f o m n s e t of generators of Z(L). Hence, oiie has (12)'
Z(L) = { x = z k O
n-r
+ zqr
n-r
kq
+
n-1
... + zq1
n-1
kq
: kEK}.
Proof.
The C o r o l l a r y f o l l o w s from t h e proof o f t h e p r e v i o u s P r o p o s i t i o n i f n-r n-r+l n- 1 one o b s e r v e s t h a t A't= A(zo,O 0,z: .z:-~ zq ). 1 t PROPOSITION 7 . If = ( z o z l . . . z ~ - ~ E) K" is a sokction of the linear system
z
(13)
A(L)y =
b, y
,....
,...,
= ( y o y1
...
=
(b bq
... bq
0
n-1
then, for every v E K w i t h t r ( v ) # 0 , (14)
x = (z v 0 0
+
z:-lvq
2
+ zq vq n-2
2
+...+
n-1 n-1 zq vq )/tr(v) 1
It,
Roots of Affine Polynomials
307
is a root of the poZynomiaZ ( 1 ) . P r o o f . Let
A(?)
-At (5)=
=
A(zo,zl,,s.,zn-l).
(2, -'zq, T 2 z-q
Raising L ( L ) z =
b
2
Then
,..., -T - ( n - l ) z q -
n- 1
).
- = Lq o r A(L)(X-'L~)
t o t h e power q , we o b t a i n A q ( L ) z q =bq=(bq bq
Using ( 4 ) , we h a v e , t h e r e f o r e , -TA(L)L1zq
=
2.
2
...bq
n-1
b)t.
Iterating this,
we f i n d
Thus, i t f o l l o w s , t A(L)A ( 2 )
(b b
=
... b).
Now, t h e r i g h t m u l t i p l i c a t i o n o f t h i s e q u a t i o n by n-1 ( t r ( v ) ) b ,I1= (v' v'q., , v f q
A ( L )1'=
=
( v vq...
n-1 vq ) t gives
lt
COROLLARY 8. The poZyizorniaZ (1) hus r o o t s i ? i K ,if and onZy if rank(A(1,)) =
rank(A(L) Ib-)
=
r . If t h i s c ondition holds, one izus IZ(P)I
= qn-r.
Proof. I f ( 1 ) h a s a r o o t x E K t h e n r a i s i n g b o t h s i d e s of t h e e q u a l i t y 0
t o t h e powers q , q
Thus,
&=
n-1
(xo
,..., q n-1 , we
... + 1:-2x: .....................
find
n- 1
= bq,
+ '10 xq0 +
l:-l~o
:1
2
xi +
n- 1
xo+ 1; X:
...
...
+ :1
n-1
n-1 n-1 xq = bq 0
n- 1 xq ) t is a s o l u t i o n of ( 1 3 ) . From h e r e and from Prop.7, 0
it
f o l l o w t h a t (1) h a s r o o t s i n K i f and o n l y i f ( 1 3 ) h a s s o l u t i o n s .
Taking i n account (8), t h e l a s t p a r t o f t h e a s s e r t i o n f o l l o w s from Prop.5.u In p a r t i c u l a r , we find the following RESULT (Dickson 1 3 ) ) .
c r n d onz$
If d e t ( A -( L ) ) #
T k ma[) L : K
-t
K, x
+
L ( x ) is n p r m u t a t i o n on K f,f
0.
Moreover, we o b s e r v e t h a t i f d e t ( A ( L ) ) # 0 , t h e o n l y r o o t x E K of t h e 0
polynomial ( 1 ) can be determined u s i n g Cramer's r u l e , t h a t i s
x 0= d e t ( b, ($,
Al.... ,&-l)
4 ,...,$-l)/det(%, =
A19 .
-
*
sS-1)
3
A(L).
I n g e n e r a l , t h e a f f i n e s u b v a r i e t y of R c o n s i s t i n g o f t h e s o l u t i o n s of polynomial ( 1 ) is given by (8) w i t h xo and Z ( L ) e x p r e s s e d by ( 1 4 ) and ( 1 2 ) ' respectively
.
308
G.Meniclietti From C o r o l l a r y 8 , we deduce t h e f o l l o w i n g u s e f u l
OBSERVATION. A polynomiaz (1) W i t h d e g ( L ( x ) )
=
q d , 0 ,< d
corripZetcZy redueible in K if and on2y if r a n k ( L ( 1 ) j b )
=
<
n-1, is
rank(A(L)) = n-d.
Another consequence i s t h e f o l l o w i n g
PROPOSITION 9 . Tuo a f f i n e poZynomiaZs, (1) and P ' ( x )
=
L'(x)
-
b ' , have
common u>oots in K if and onZy ?'f t h e equalions of t h e Zinear sistems (13) and A(L') -
y= b ' m e compatible. Proof. If x E K i s acommon r o o t of b o t h P(x) and P'(x) t h e n x
(xo :x
n-1
... x:
4
O
=
) t i s a s o l u t i o n f o r both l i n e a r systems i n t h e a s s e r t i o n .
Conversely, i f t h e e q u a t i o n s of b o t h systems a r e c o m p a t i b l e , we f i n d , by ( 1 4 ) , a common r o o t f o r t h e given polynomials.[ I t i s easy t o prove t h a t , when t h e c o n d i t i o n of t h e p r e v i o u s p r o p o s i t i o n i s
s a t i s f i e d , t h e s e t of common r o o t s f o r P(x) and P ' ( x ) i s an a f f i n e s u b v a r i e t y of
,-;I
K whose dimension i s n - r ' ,
(-Gi1:
A(L)
r ' = rank
-
where =
(-:
A(L)
)
b
-I - :. A(L')I b'
rank
--
Now we want t o use t h e p r e v i o u s r e s u l t s t o d i s c u s s t h e e q u a t i o n
xq
(15)
m
- x
=
b , b E K , 1 ,< m , < n-1
.
F i r s t we observe t h a t , given d = (n,m) and k = n / d , t h e i n t e g e r s i m + j ,
i
= O,l,
...,k-1,
j = O , l , . . .,d-1,
a r e p a i r w i s e incongruent modulo n.
I n t h i s c a s e , t h e l i n e a r s y s t e m (13) becomes
(16)
Y2m+ j
= bq'
- 'm+j
...................
'( k-1 ) m+ j
-
(k-2 )m+j
'i
-
(k-1 )m+ j
'+m
=
'+(k-2)m bqJ
=
bq
j + (k-1 ) m
,
j = 0.1,
and t h u s i t s e q u a t i o n s a r e compatible i f and o n l y i f
...,d-1, im j k-1 j+im k-1 C bq = ( C bq )' = 0. i=O
i=O
From t h i s , we deduce t h a t (15) has some r o o t s i n K i f and o n l y i f (17)
k-1 im C bq = t r F , ( b ) = 0,
i =O
d where F' = GF(q )
(')
C_
GF(qn)
(2),
The i n t e g e r s h d , h = 0 , 1 ,
modulo m and t h e r e f o r e
k-1
I: bq
i =O
... ,k-1, im
k-1 =
and i m , i = 0,1,
C bq h=O
hd
.
...,k-1,
a r e congruent
Roots of Affine Polynomials m
-
L(x) = xq
309
x implies obviously
d Z ( L ) = GF(q ) , d = ( n , m ) .
(18)
T h e r e f o r e , w e can d e t e r m i n e a r o o t x E K of (15) u s i n g P r o p . 7 and supposing t h a t C
(17) i s s a t i s f i e d .
From (16), by s u c c e s s i v e s u b s t i t u t i o n s , we f i n d yim+l
i-1
= yj + (
C bq
h=O
and by (17) yim+j =
A. 1
hm
k-1
C bq
(
hm
h=i
)q
)'
j
j
, i
1,2
=
, X.EK,
,...,k-1,
i = 0,1,
J
...,d-1,
j = 0,1,
...,k-1,
j = O,l,...,d-l
Let u s c o n s i d e r t h e p a r t i c u l a r s o l u t i o n =
'im+j
X
obtained f o r
j
-
(
k-1
C bq
hm
h=i
)q
j
, i
O,l,
=
...,k-1,
j
=
...,d-1.
= 0, j = 0,1,
From ( 1 4 ) w e o b t a i n
x tr(v) 0
n-1 =
h
E 24 vq n-h
h=O d- 1
Hence, s e t t i n g v = wq
, we
h
'
k-ld-1 =
i=O
qn-(im+j)
j=o 'im+j
n-(im+j)
"4
have
k-1 d-1 n-(im+j) n-(im+j)+d-1 C C z;m+j wq i=O j=O where t r ( w ) = t r ( v ) # 0.
x tr(w) 0
=
I f we o b s e r v e t h a t n-(im+j) = -
zSm+j
rm k-1 n+(h-i)m k-i-1 Cbq = C bq , h=i r=O
t h e n , s u b s t i t u t i n g i n t o t h e p r e v i o u s e q u a l i t y , one h a s
x tr(w) 0
rm d-1 n-im+(d-1-j) k-1 k-i-1 Z C bq C wq i = O r=O j =O k-1 k-i-1 r m d-1 n-im+s = - C C bq C w q i = O r=O s =o k-1 k-i-1 rm d-1 s n-im = - C Z bq ( C w ' ) ~ i=O r=O s=o =
-
.
From h e r e , p u t t i n g a =
d-1 C
s W
~
s=o
and o b s e r v i n g
tr(w) =
we deduce
k-1 d-1
c
i=o
I: wq
j=O
im+j
k-1 =
c
i=O
a'
im
= trF,(a),
...,d-1,
0,1,
.
G.Menichetti
310 k-1 C i=O k = C h=1 k-1
xOtrFI(a) =
-
= -
c
k-i-1 rm n-im C bq aq r=O r m hm h-1 C b q aq
r=O
h-1 ~
h = l r=O
b
rm
q aq
hm
.
T h e r e f o r e : The equation (15) has r o o t s i n K = GF(qn) i f and only i f b s a t i s f i e s
the condition ( 1 7 ) . If such condition i s s a t i s f i e d , the s e t of r o o t s i s the a f f i n e subvariety ( 8 ) in which Z ( L ) i s given by (18) and x
=--
k-1 h-1 rm hm C C b q a' trF,(a) h=l r=O
,
t r F l ( a ) # 0.
I f ( k , p ) = 1 ( p = c h a r K ) then t r F l ( l ) = k # 0 and t h e r e f o r e , we can s e t a = l . The p r e v i o u s r e s u l t a l l o w s u s t o determine t h e r o o t s of a second d e g r e e e q u a t i o n i n a f i e l d K of c h a r 2 . I n f a c t , f o r q = 2 , m = 1, w e f i n d t h e w e l l known c o n d i t i o n t r ( b ) = 0 i n o r d e r t h a t t h e e q u a t i o n X'
+ x
t
b = 0
h a s a r o o t in K = GF(2").
Moreover, from (18) and ( 1 9 ) , we deduce t h a t t h e r o o t s
of t h e above e q u a t i o n a r e x
n-1 =--
C
h-1
C b
t r ( a ) h=l r=O
2r 2h a
and
xo+ 1 ,
where a E K i s a f i x e d element w i t h t r ( a ) # 0.
REFERENCES
[ l ] Berlekamp, E . R . ,
AZgebraic coding theory (Mc Graw Book Company,New York,1968).
121 B i l i o t t i M. and M e n i c h e t t i G . , On a g e n e r a l i z a t i o n of Kantor's l i k e a b l e planes, Geom. D e d i c a t a , 1 7 (1985) 253-277.
[ 3 ] Dickson, L . E . ,
Linear Groups w i t h an e x p o s i t i o n o f t h e Galois fieZd theory
(Teubner, L e i p z i g . R e p r i n t Dover, New York, 1958).