On Binomial Ideals associated to Linear Codes

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by  N. Dück

On Binomial Ideals Associated to Linear Codes
Natalia Du¨ck, Karl-Heinz Zimmermann
January 14, 2014
4
1
0 Abstract
2
Recently,it was shown that a binary linear code can be associated to

n a binomial ideal given as the sum of a toric ideal and a non-prime ideal.
a Sincethentwodifferentgeneralizationshavebeenprovidedwhichcoincide
J
forthebinarycase. Inthispaper,weestablishsomeconnectionsbetween

3 the two approaches. In particular, we show that the corresponding code
1
idealsarerelatedbyelimination. Finally,anewheuristicdecodingmethod

for linear codes overprime fields is discussed using Gr¨obner bases.
]
C
A 1 Introduction
.
h
t Digital data are exposed to errors when transmitted through a noisy channel.
a
But since receiving correct data is indispensable in many applications, error-
m
correctingcodesareemployedtotacklethisproblem. Byaddingredundancyto
[
the messages, errors can be detected and corrected [13, 21].

1 Gr¨obner bases, on the other hand, are a powerful tool that has originated
v from commutative algebra providing a uniform approachto graspa wide range
4
of problems such as solving algebraic systems of equations, testing ideal mem-
9
bership, and effective computing in residue class rings modulo polynomial ide-
7
2 als [1, 8].
. In [5] both subjects have been linked by associating a binary linear code to
1
0 a certainbinomialidealgivenasthe sumofa toricidealanda non-prime ideal.
4 In addition, the authors demonstrated how the computation of the minimum
1 distance can be accomplished by a Gr¨obner basis computation. In [17, 18] this
v: approach has been extended to codes over finite prime fields, whose associated
i binomial ideals are called code ideals. In this way, several concepts from the
X
rich theory of toric ideals can be translated into the setting of code ideals.
r
Another generalizationof [5] was given in [12, 15, 19]. The approachin [15]
a
coversthe generalcaseof linear codes overarbitraryfinite fields by introducing
the so-called generalized code ideal.
This paper pursues two objectives. First, both approaches are linked to
provide further inside into the structure of ideals associated to linear codes.
In the binary case both approaches are the same and it will be shown that in
the caseofa prime fieldthe code idealis aneliminationidealofthe generalized
codeideal. Furthermore,itwillbeprovedthatthereducedGr¨obnerbasisforthe
generalized code ideal with respect to the lexicographic order can be explicitly
constructed from a generator matrix in standard form.
Second,aheuristicmethodisintroducedwhichallowstodecodelinearcodes
using the corresponding code ideal instead of the generalized code ideal. One
1
of the main reasonsfor introducing the generalizedcode ideal is that it enables
us to apply the same procedure for decoding as in the binary case, but at the
expense of introducing considerably more variables. Since for codes over prime
fields the code ideal provides an alternative to the generalizedcode ideal which
requiresonlya (q−1)-fractionalamountofvariablesit is reasonableto lookfor
an alternative way of decoding based on the code ideal. We will provide such a
heuristic method and discuss its pros and cons.
2 Preliminaries
2.1 Linear Codes
Let F denote the finite field with q elements. A linear code C of length n and
q
dimensionk overF isthe imageofa one-to-onelinearmapping fromFk to Fn.
q q q
In other words, the code C is a subspace of the vector space Fn of dimension
q
k ≤n. SuchacodeC iscalledan[n,k]codewhoseelementsarecalledcodewords
and are usually written as row vectors.
Ageneratormatrix foran[n,k]codeC isak×nmatrixGoverF whoserows
q
form a basis of C. A generator matrix in reduced echelon form G = (I |M),
k
where I denotes the k×k identity matrix, is said to be in standard form and
k
the corresponding code C is called systematic.
A parity check matrix H for an [n,k] code C is an (n−k)×n matrix over
F suchthat a wordc∈Fn belongs to C if and only if cHT =0. It follows that
q q
the code C equals the kernel of the matrix H given as a mapping from Fn to
q
Fn−k [13, 21].
q
2.2 Gr¨obner Bases and Toric Ideals
Write K[x] = K[x ,...,x ] for the commutative polynomial ring in n inde-
1 n
terminates over an arbitrary field K and denote the monomials in K[x] by
xu =xu11xu22···xunn, where u=(u1,...,un)∈Nn0.
A monomial order on K[x] is a relation ≻ on the set of monomials in K[x]
(or equivalently, on the exponent vectors in Nn) satisfying: (1) ≻ is a total
0
ordering, (2) the zero vector 0 is the unique minimal element, and (3) u ≻ v
implies u+w ≻v+w for all u,v,w∈Nn.
0
Givenamonomialorder≻,eachnon-zeropolynomialf ∈K[x]hasaunique
leading term, denoted by lt (f), which is given by the largest involved term.
≻
The leading ideal of an ideal I w.r.t. a monomial order ≻ is the monomial
ideal generated by the leading monomials of its elements,
lt (I)=hlt (f)|f ∈Ii. (1)
≻ ≻
The Gr¨obner basis for an ideal I in K[x] w.r.t. ≻ is a finite subset G of I with
thepropertythattheleadingtermsofthepolynomialsinG generatetheleading
ideal of I, i.e.,
lt (I)=hlt (g)|g ∈Gi. (2)
≻ ≻
Amonomialxα ∈/ lt (I)is calledastandard monomial. Thesetofallstandard
≻
monomials forms a basis for the K-algebra K[x]/I.
If no monomial in a Gr¨obner basis is redundant and for any two distinct
elements g,h ∈ G, no term of h is divisible by lt (g), then G is called reduced.
≻
A reduced Gr¨obner basis is uniquely determined (provided that the generators
are monic) and henceforth the reduced Gr¨obner basis for an ideal I w.r.t. ≻
will be denoted by G (I). For more information on Gr¨obner basics the reader
≻
should consult [1, 2, 8].
A binomial in K[x] is a polynomial consisting of two terms, i.e., a binomial
is of the form c xα −c xβ, where α,β ∈ Nn and c ,c ∈ K are non-zero. A
α β 0 α β
binomial is pure if the involved monomials are relatively prime. A binomial
ideal is an ideal generated by binomials.
Let A = (a ) be a non-negative integral m×n matrix and take the poly-
ij
nomial rings K[x] = K[x ,...,x ] and K[y] = K[y ,...,y ]. Define the K-
1 n 1 m
algebra homomorphism ϕ : K[x] → K[y] by ϕ(xi) = y1a1iy2a2i · ymami, where
a =(a ,a ,...,a )T denotes the ith column of the matrix A for 1≤i≤n.
i 1i 2i mi
The kernel of the morphism ϕ is an ideal of K[x], called toric ideal associated
to the matrix A, and denoted by I =ker(ϕ).
A
For any integer u, write u+ = max{0,u} and u− = (−u)+ and for any
integer vector u = (u ,...,u ) define the corresponding vectors u+ and u−
1 n
componentwise. Clearly, the vectors u+ and u− have disjoint support and thus
any vector u ∈ Zn can be uniquely written as u = u+ −u−. For instance, if
u = (2,0,−3), then u+ = (2,0,0) and u− = (0,0,3). In view of this notation,
the toric ideal I is generated by pure binomials [4, 20],
A
IA = xu+ −xu− |u=u+−u− ∈kerZ(A) , (3)
D E
where kerZ(A) denotes the kernel of the matrix A defined as a mapping from
Zn to Zm.
3 The Code Ideal
LetC bean[n,k]codeoverafinitefieldF ,wherepisaprimenumber,andletK
p
beanarbitraryfield. Inviewof[18],definethecodeideal inK[x]=K[x ,...,x ]
1 n
associated to the code C as a sum of binomial ideals
I(C)=I′(C)+I (4)
p
where
I′(C)=hxc−xc′ |c−c′ ∈Ci (5)
and
I =hxp−1|1≤i≤ni. (6)
p i
In terms of the ideal I , the exponent of any monomial can be treated as a
p
vector in Fn because for any 1≤i≤n and 0≤r ≤p−1,
p
xp+r ≡xp+r −xr ·(xp−1)=xr mod I
i i i i i p
and thus by induction for any integer m≥0,
xm·p+r ≡xr mod I .
i i p
The code ideal I(C) can be based on a toric ideal. To see this, let H be a
parity check matrix for the code C and let H′ be an integral matrix such that
H =H′⊗ZFp. Then
I(C)=I +I . (7)
H′ p
It follows thatthe code idealis the sumof a prime idealanda non-prime ideal.
Although the code ideal is not toric, it resembles a toric ideal in some respects.
Similar to (3) the code ideal is generated by pure binomials xu −xu′, where
u−u′ belongs to the kernel of H.
The binomial xu−xu′ ∈I(C) is said to correspond to the codeword u−u′.
However, note that in contrast to the integral case, there is no unique way of
writing u = u+ −u−. For example, the word (1,1,0) in F3 can be written as
2
(1,1,0) = (0,1,0)−(1,0,0) or (1,1,0) = (1,0,0)−(0,1,0). Thus, different
binomials may correspond to the same codeword.
Note that the reduced Gr¨obner basis w.r.t. the lexicographic (lex) ordering
with x ≻ x ≻ ... ≻ x can be directly read off from a generator matrix in
1 2 n
standard form [17]. More specifically, let G be a standard generator matrix for
C withrowvectorsg =e −m ,wheree denotestheithunitvector,1≤i≤k.
i i i i
Then the reduced Gr¨obner basis G (I(C)) w.r.t. the lex ordering is given by
≻
G≻(I(C))={xi−xmi |1≤i≤k}∪{xpi −1|k+1≤i≤n}. (8)
Givenan[n,k]code C overF . The correspondingcode idealcanbe consid-
p
eredasaneliminationidealofatoricideal[14,Remark1]. Thiswillbespecified
in Prop. 3.1. Beforehand, we require further definitions.
Toanon-negativeintegralm×nmatrixAassociatetheintegralm×(m+n)
matrix
A(p)=(A|p·I ). (9)
m
For an integral matrix A ∈ Zm×n, let kerZ(A) denote the kernel of A as a
mapping from Zn to Zm, and let kerp(A) denote the kernel of the matrix A⊗Z
Z as a mapping from Zn to Zm. Note that for any vector u ∈ Zn, u ∈
p p p p
ker (A) is equivalent to Au ≡ 0 mod p or Au = pv for some v ∈ Zm, which
p
in turn is equivalent to (u,−v)∈kerZ(A(p)) for some v ∈Zm. In other words,
thereisabijectivecorrespondencebetweenkerZ(A(p))andkerp(A)givenbythe
projection onto the first n coordinates.
In the following, the toric ideal associated to the matrix A(p) is studied in
the polynomial ring K[x,y]=K[x ,...,x ,y ,...,y ].
1 n 1 m
Proposition 3.1. Let I(C) be the code ideal of an [n,k] code C over F with
p
parity check matrix H and let I be the toric ideal associated to the non-
H′(p)
negative integral matrix H′ with H =H′⊗ZFp. The code ideal I(C) is given as
elimination ideal
I(C)= I +hy −1,...,y −1i ∩K[x]. (10)
H′(p) 1 m
Equivalently, (cid:0) (cid:1)
I(C)= f(x,1)|f ∈I , (11)
H′(p)
where 1 denotes the all-1 vector.(cid:8) (cid:9)
Proof. It is clear that the statements (10) and (11) are equivalent. Thus it is
sufficient to prove that (11) holds.
Let xa−xb ∈I(C) and so a−b∈ker (H′). By the preceding remark,there
p
is a vector d ∈ Zm such that (a−b,d) ∈ kerZ(H′(p)) and so xayd+ −xbyd−
belongs to I . Conversely, let xaya′ −xbyb′ ∈ I . Then (a−b,a′−b′)
H′(p) H′(p)
belongstokerZ(H′(p))andbythebijectivecorrespondencebetweenkerZ(H′(p))
and ker (H′), a−b belongs to C and thus xa−xb ∈I(C).
p
Example 1. Consider the [3,2] code C over F with generator matrix
7
1 0 4
G=
0 1 1
(cid:18) (cid:19)
andcorrespondingparitycheckmatrixH = 1 2 5 . ChooseH′ = 1 2 5
and so H′(7) = 1 2 5 7 . A computation in Singular [10] provides the
(cid:0) (cid:1) (cid:0) (cid:1)
reduced Gr¨obner basis for I w.r.t. the lex ordering,
(cid:0) H′(cid:1)(7)
G ={x7−y5,x5−x2,x y4−x6,x2y3−x5,x3y2−x4,
3 2 3 2 3 2 3 2 3
x4y−x3,x x −y,x2−x ,x y−x4,x x −x3,x x2−x ,}.
2 3 2 3 1 2 1 2 1 3 2 1 2 3
Substituting y =1 for all these binomials yields the set
{x7−1,x5−x2,x −x6,x2−x5,x3−x4,x4−x3,
3 2 3 2 3 2 3 2 3 2 3
x x −1,x2−x ,x −x4,x x −x3,x x2−x }.
2 3 1 2 1 2 1 3 2 1 2 3
The reduced Gr¨obner basis for the ideal generated by these polynomials is
{x −x3,x −x6,x7−1},
1 3 2 3 3
which coincides with the reduced Gr¨obner basis for I(C) as given in (8). ♦
The matrix H′ in Prop. 3.1 can always be chosen to be non-negative. In this
way, working with Laurent polynomials or an additional indeterminate can be
avoided.
4 The Generalized Code Ideal
In the preceding section, the code ideal associated to a linear code over a finite
prime field has been introduced. Now the code ideal corresponding to a linear
code over an arbitrary finite field is described following [15].
For this, let C be an [n,k] code over the field F , where q = pr, p is a
q
prime, and r ≥ 1 is an integer. Let α be a primitive element of F , i.e., F =
q q
0,α,α2,...,αq−2,αq−1 =1 . The crossing map
(cid:8) (cid:9) N:Fn →Zn(q−1)
q
is defined as
a=(a1,...,an)=(αj1,...,αjn)7→(ej1,...,ejn),
where e is the ith unit vector of length q−1, 1 ≤ i ≤ q −1, and each zero
i
coordinateismappedtothezerovectoroflengthq−1. Theassociatedmapping
H:Zn(q−1) →Fn
q
is given as
q−1 q−1
(j ,...,j ,j ,...,j )7→ j αi,..., j αi .
1,1 1,q−1 2,1 n,q−1 1,i n,i
!
i=1 i=1
X X
For instance, in view of the field F ={0,α=2,α2 =4,α3 =3,α4 =1},
5
N(1,0,3)=N(α4,0,α3)=(e ,0,e )=(0,0,0,1,0,0,0,0,0,0,1,0)
4 3
and
H(0,0,0,1,0,0,0,0,0,0,1,0)=(α4,0,α3).
Note that the mapping H is the left inverse of the crossing map N, i.e., H◦N is
the identity on Fn, but it is not the right inverse.
q
Put x =(x ,x ,...,x ), 1≤j ≤n, and x=(x ,...,x ). Define the
j j1 j2 j,q−1 1 n
generalized code ideal associated to the code C as
I (C)= xNa−xNb |a−b∈C ⊆K[x]. (12)
+
Forinstance,inviewofthep(cid:10)reviousexample,xN(1,(cid:11)0,3) =x(0,0,0,1,0,0,0,0,0,0,1,0) =
x x .
14 33
AgeneratingsetforthecodeidealI (C)willcontainbothageneratingsetof
+
the associatedlinear code and the associated scalar multiples, and an encoding
of the additive structure of the field F [15, 19]. The latter can be givenby the
q
ideal I in K[x] generated by the set
q
n
({x x −x |αu+αv =αw}∪{x x −1|αu+αv =0}). (13)
iu iv iw iu iv
i=1
[
Theorem 4.1 ([15]). Let C be an [n,k] code over F and suppose g ,...,g
q 1 k
are the row vectors of a generator matrix for C. The generalized code ideal
associated to the code C is
I (C)=I +I , (14)
+ G q
where I is an ideal of K[x] with generating set
G
xN(αjgi)−1|1≤i≤k,1≤j ≤q−1 . (15)
n o
Note that the binomials in I are squarefree. The next result exhibits the
G
type of binomials which belong to a generalized code ideal.
Lemma 4.2. Let C be an [n,k] code over F and let xa−xb be a binomial in
q
K[x]. If H(a−b)∈C, then xa−xb ∈I (C).
+
Proof. Put a′ = Ha and b′ = Hb. Since the mapping H is linear, a′ −b′ =
Ha−Hb=H(a−b). Supposea′−b′ ∈C. Thenbydefinition,xNa′−xNb′ ∈I (C).
+
Claim that xa−xb ≡xNa′ −xNb′ mod I . Indeed, write
q
n
xa = xai1xai2···xai,q−1.
i1 i2 i,q−1
i=1
Y
Then the ith entry of the word a′ is a α + a α2 + ··· + a αq−1 = β ,
i1 i2 i,q−1 i
where either βi = 0 or βi = αℓi for some 1 ≤ ℓi ≤ q − 1. It follows that
xai1xai2···xai,q−1 ≡ 1 mod I or xai1xai2···xai,q−1 ≡ x mod I . Thus
i1 i2 i,q−1 q i1 i2 i,q−1 iℓi q
xNa′ = n x and so xa ≡ xNa′ mod I . Applying the same argument
i=1 iℓi q
βi6=0
to xb estQablishes the claim and so the assertion.
Note that the mapping N is not linear, since e.g. over F ,
5
N(1,3)+N(1,1)=N(α4,α3)+N(α4,α4)=(e ,e )+(e ,e )
4 3 4 4
and
N((1,3)+(1,1))=N(2,4)=N(α,α2)=(e ,e ).
1 2
However, the operator N applied to the exponent of a monomial is quasi-
linear as described in the following.
Lemma 4.3. For any vectors a,b in Fn,
q
xNa+Nb ≡xN(a+b) mod I .
q
Proof. Let a = αi1,...,αin and b = αj1,...,αjn . Assume that all entries
in a and b are non-zero; the more general case can be similarly handled. Put
(cid:0) (cid:1) (cid:0) (cid:1)
a+b= αk1,...,αkn and assume that the zero entries are at the positions in
the set J ⊆ {1,...,n}, i.e., αis +αjs = αks for s ∈/ J and αis +αjs = 0 for
(cid:0) (cid:1)
s ∈ J. The binomials x x −x for s ∈/ J and x x −1 for s ∈ J
s,is s,js s,ks s,is s,js
belong to I . Moreover,
q
n n
xNa+Nb = xeis+ejs = x x
s s,is s,js
s=1 s=1
Y Y
and
xN(a+b) = xeks = x .
s s,ks
s∈YJ\n s∈YJ\n
Butx x ≡x mod I fors∈J\nandx x ≡1 mod I fors∈J.
s,is s,js s,ks q s,is s,js q
By comparing both equations, the result follows.
Note that this result has been implicitly used in [15, Theorem 2.1].
4.1 The generalized code ideal for codes over prime fields
For a binary [n,k] code C, the generalized code ideal equals the code ideal.
To see this, note that F = {0,α = 1} and the generalized code ideal can be
2
considered as an ideal in K[x ,...,x ] instead of K[x ,...,x ].
1 n 11 n1
Moreover, if G is a generator matrix for C with rows g ,...,g , then the
1 k
code ideal I(C) has the generating set [14, Theorem 3.2]
{xgi −1|1≤i≤k}∪ x2−1|1≤i≤n .
i
By Thm. 4.1, the generalized code ideal(cid:8)I (C) has the same(cid:9)generating set.
+
Now let C be an [n,k] code over a finite field F , where p > 2 is a prime.
p
Recallthatx=(x ,...,x )andx =(x ,...,x )for1≤j ≤n. Moreover,
1 n j j1 j,p−1
put x =(x ,...,x ) for 1≤i≤q−1. The generalized code ideal belongs to
i 1i ni
the ring K[x]=K[x ,...,x ] whereas the code ideal
11 n,p−1
I(C)= xa−xb |a−b∈C (16)
i i
can be considered to belong to K(cid:10)[x ]⊂K[x] for any(cid:11)1≤i≤p−1.
i
Proposition 4.4. Let C be a linear code of length n over a prime field F . The
p
code ideal I(C) as defined in (16) is an elimination ideal of the ideal I (C) as
+
defined in (12). More precisely,
I(C)=I (C)∩K[x ] for each 1≤i≤q−1.
+ i
Proof. Let xa − xb ∈ I(C), i.e., a − b ∈ C. Clearly, xa − xb = xa′ − xb′,
i i i i
wherea′ =(a e ,...,a e )andb′ =(b e ,...,b e ). Furthermore, H(a′−b′)=
1 i n i 1 i n i
αi(a−b)∈C and so by Lem. 4.2, xa−xb ∈I (C)∩K[x ].
i i + i
Conversely, let xa−xb be a binomial in I (C)∩K[x ]. Clearly, a−b must
+ i
be of the form ((a −b )e ,(a −b )e ,...,(a −b )e ) with
1 1 i 2 2 i n n i
H(a−b)=αi·(a−b)∈C.
ButasC islinear,αp−i−1(αi(a−b))=a−b∈C andsobyconvention,xa−xb =
xa−xb ∈I(C).
i i
Example 2. Consider the ternary [6,3] code C generated by the matrix
1 0 0 2 2 0
G= 0 1 0 1 1 0 .
 
0 0 1 1 2 1
 
By Thm. 4.1, the generalized code ideal I (C) has the generators
+
x x x −1, x x x −1, x x x −1,
12 41 51 11 42 52 22 42 52
x x x −1, x x x x −1, x x x x −1
21 41 51 32 42 51 62 31 41 52 61
and
x2 −x , x x −1, x2 −x , 1≤i≤6.
i1 i2 i1 i2 i2 i1
Computations in Singular[10] show that the elimination ideal
I (C)∩K[x ,x ,x ,x ,x ,x ]
+ 11 21 31 41 51 61
is generated by the binomials
x3 −1, x3 −1, x3 −1, x −x2 x x2 , x −x2 x2 , x −x x .
61 51 41 31 41 51 61 21 41 51 11 41 51
Similarly, the elimination ideal
I (C)∩K[x ,x ,x ,x ,x ,x ]
+ 12 22 32 42 52 62
is generated by
x3 −1, x3 −1, x3 −1, x −x2 x x2 , x −x2 x2 , x −x x .
62 52 42 32 42 52 62 22 42 52 12 42 52
Comparing these generators with the reduced Gr¨obner basis for the code ideal
I(C) given in (8) confirms that both elimination ideals are (up to renaming of
variables) equal to I(C). ♦
Next we show that the reduced Gr¨obner basis for a generalized code ideal
w.r.t. the lex ordering can be easily constructed from a standard generator
matrix.
Theorem 4.5. Let C be an [n,k] code over a prime field F with primitive
p
element α and let g ,...,g be the row vectors of a generator matrix for C in
1 k
standard form. The reduced Gr¨obner basis for the generalized code ideal I (C)
+
w.r.t. the lex ordering x ≻x ≻...≻x is given by
11 12 n,p−1
m(j)
G = x −x i |1≤i≤k, 1≤j ≤p−1 (17)
ij p−1
(cid:26) (cid:27)
∪ x −xαj |k+1≤i≤n, 1≤j ≤p−2 (18)
ij i,p−1
∪nxp −1|k+1≤i≤n o (19)
i,p−1
where (cid:8) (cid:9)
m(j) =(e −g )αj, 1≤i≤k, 1≤j ≤p−1. (20)
i i i
Proof. Note that the support of each vector m(j) lies in {k+1,...,n}. Thus
i
m(j)
themonomialx i involvesonlythevariablesx ,x ,...,x . It
p−1 k+1,p−1 k+2,p−1 n,p−1
followsthatthesecondtermsdonotinvolveanyoftheleadingterms. Moreover,
different binomials in G have relatively prime leadings terms. Hence, G is the
reduced Gr¨obner basis w.r.t. lex ordering for the ideal it generates.
It remains to show that G generates the ideal I (C). First, claim that G ⊂
+
I (C). Indeed, consider the following cases:
+
• Takeabinomialx −xmi(j) fromthesubset(17). ApplyingthemappingH
ij p−1
to the exponents of the involved monomials yields αje and αp−1m(j) =
i i
m(j). But αje −m(j) = αjg ∈ C and so by Lem. 4.2 the considered
i i i i
binomial belongs to I (C).
+
• Consider a binomial x − xαj from the subset (18). Applying the
ij i,p−1
mapping H to the exponents of the monomials in this binomial gives
αje − αp−1αje = 0 ∈ C and thus by Lem. 4.2 this binomial lies in
i i
I (C).
+
• Pick a binomial xp −1 from the subset (19). It obviously corresponds
i,p−1
to the zero word and therefore belongs to I (C).
+
Second, claim that I (C)⊂hGi. Indeed, put
+
J = x −xαj |k+1≤i≤n, 1≤j ≤p−2
ij i,p−1
D E
and
K = xp −1|k+1≤i≤n .
i,p−1
Consider the following cases:
(cid:10) (cid:11)
• Firstwe provethat the binomials in I are generatedby the binomials in
G
G: For this, consider the binomial xN(αjgi) −1 for some 1 ≤ i ≤ k and
1≤j ≤p−1. By definition, αjg =αje −m(j). Claim that
i i i
xN(αjgi)−1≡xN(cid:16)−mi(j)(cid:17) x −xmi(j) mod J.
ij p−1
(cid:18) (cid:19)
Indeed,
xN(αjgi) =xN(αjei)xN(cid:16)−mi(j)(cid:17) =x xN(cid:16)−mi(j)(cid:17).
ij
Moreover, the squarefree monomial xN(cid:16)−mi(j)(cid:17) has supp m(j) ⊆ {k +
i
1,...,n} and so only involves the variables xk+1,...,xn.(cid:16)If th(cid:17)e variable
x forsomek+1≤s≤nand1≤t≤p−1isinvolvedinxN(cid:16)−mi(j)(cid:17),then
st
the s-th coordinate of m(j), say αm, is non-zero and satisfies −αm = αt.
i
Hence, the monomial xmi(j) contains the variable xαm .
p−1 s,p−1
Two cases occur: If 1≤t≤p−2, then x −xαt ∈G and thus
st s,p−1
x xαm ≡xαt xαm =xαt+αm ≡x0 =1 mod J +K.
st s,p−1 s,p−1 s,p−1 s,p−1 s,p−1
Otherwise, t=p−1 and then
x xαm =xαm+1 ≡x0 =1 mod J +K.
s,p−1 s,p−1 s,p−1 s,p−1
Therefore, both cases provide xN(cid:16)−mi(p−1)(cid:17)xmi(p−1) ≡1 mod J +K.
p−1
• Second we prove that the binomials in I whose second term is unequal
q
to 1 are generatedby the binomials in G. For this, let αu+αv =αw with
αu,αv,αw 6=0 and consider the following cases:
– Let 1≤i≤k. We show that x x −x ∈hGi. Take the following
iu iv iw
polynomial which obviously belongs to hGi,
m(w) m(u) m(u) m(v)
x −x i −x x −x i −x i x −x i
iw p−1 iv iu p−1 p−1 iv p−1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
m(w) m(u) m(v)
=x −x x − x i −x i x i
iw iu iv p−1 p−1 p−1
(cid:18) (cid:19)
≡x −x x mod K,
iw iu iv
where the last step follows from
xmi(u)xmi(v) =x(ei−gi)(αu+αv) ≡x(ei−gi)αw =xm(iw) mod K. (21)
p−1 p−1 p−1 p−1 p−1