MM Research Preprints, 82–88
KLMM, AMSS, Academia Sinica
No. 24, December 2005
On Partial Difference Coherent and Regular
Ascending Chains
Xiao-Shan Gao and Gui-Lin Zhang
Key Laboratory of Mathematics Mechanization
Institute of Systems Science, AMSS, Academia Sinica
1. Introduction
In this paper, we propose a characteristic set theory for partial difference polynomial
systems. First, we introduce the concept of coherent ascending chains and prove that the
difference polynomials in the saturation ideal of a coherent ascending chain has a canonical
representation. Second, we introduce the concept of regular chains and prove that a partial
difference ascending chain is the characteristic set of its saturation ideal if and only if it is
coherent and regular. We also prove that the saturation ideal of a partial difference regular
and coherent ascending chain is the union of some algebraic saturation ideals. These results
are generalizations of similar results about ordinary difference systems proposed in [6].
For the general theory of difference algebra, please refer to [4, 5, 2, 7, 9].
2. Preliminaries
We will introduce the notions and preliminary properties needed in this paper.
Let K be an inversive partial difference field of characteristic zero and {σ1 , · · · , σm } a
om of
finite set of transforming operators over K. The order of an element η = σ1o1 · · · σm
o1
o
m
m
Tσ = {σ1 · · · σm |j = 1, · · · , m, oj ≥ 0} is ord(η) = Σj=1 oj . η is proper if ord(η) 6= 0. The
im . For η , η ∈ T , η is a
vector of an element η ∈ Tσ is vec(η) = (i1 , · · · , im ) if η = σ1i1 · · · σm
1 2
σ
1
(proper) multiple transform of η2 or η2 is a (proper) factor transform of η1 if ∃ η ∈ Tσ such
that η1 = ηη2 (ord(η) 6= 0). It is denoted by vec(η1 ) − vec(η2 ) º 0(Â 0).
Let X = {x1 , · · · , xn } be a finite set of difference indeterminates over K and Tσ X =
{ηxi |η ∈ Tσ , i = 1, · · · , n}. K{X} = K{x1 , · · · , xn } = K[Tσ X] denotes the partial difference
ring of partial difference polynomials (pr-pol) in the indeterminates Tσ X with coefficients in
K. Let < be a tall ordering over Tσ X defined as follows: ∀η, θ ∈ Tσ , 1 ≤ i, j ≤ m, ηxi > θxj if
i > j or i = j and ord(η) > ord(θ) or else the first nonzero element of vec(η)−vec(θ) is greater
than zero. Let f be a pr-pol not in K, the leader of f is the highest element of Tσ X(w.r.t <)
that appears in f , and we denote it by uf . We write f = Id udf + · · · + I0 . Id = init(f ) is
the initial of f . Let f ∈ K{X} and uf = ηxi . Then i and xi are called the class and leading
variable of f , denoted as class(f ) and lvar(f ) respectively. We define vec(ηxi ) = vec(η) and
vec(f, xj ) = vec(η), if ηxj = max{τ xj appears in f }, vec(f ) = vec(f, lvar(f )).
Let g be a pr-pol, not in K. A pr-pol f is less than g if uf < ug or (uf = ug ) = u and
deg(f, u) < deg(g, u). If neither f < g nor g < f , we say that f and g are equivalent and we
write f ≡ g. A pr-pol f is partial reduced with respect to g if deg(f, ηug ) < deg(g, ug ), ∀η ∈
Tσ .
CS Method for Difference Polynomial Systems
83
A subset A of R\K, where every element is reduced with respect to all the others, is
called a partial difference ascending chain, or simply a chain.
Lemma 2.1 Every chain A of R = K{x1 , · · · , xn } is a finite set.
Proof: It is sufficient to show that for every 1 ≤ i ≤ n, the pr-pols with a given class i is
finite. It is obvious that the leaders in the ascending chain are pairwise different, so we need
only to show Λ = {ηxi |ηxi = uA , A ∈ A, class(A) = i} is a finite set. Otherwise, by Dickson’s
Lemma, there are finite η, · · · , ρ ∈ Λ such that every element of Λ is a multiple transform
of η s · · · ρt xi for some s, · · · , t. Without loss of generality, we may assume that there is an
infinite subset Λ1 ⊆ Λ every element of which is a proper multiple transform of ηxi . Continue
the above process for Λ1 , we obtain η1 . Then are infinite sequences η, η1 , η2 , · · · . Suppose
ηj xi = uAj , we have deg(A1 , uA1 ) > deg(A2 , uA2 ) > · · · , a contradiction.
If A = {A1 , · · · , Ap } is a chain with A1 < · · · < Ap , then we denote A as a sequence of
pr-pols A = A1 , · · · , Ap . If A = A1 , · · · , Ap and B = B1 , · · · , Bq are two chains, we say that
A < B if either there is some j ≤ min(p, q) such that Ai ≡ Bi for i < j and Aj < Bj , or
q < p and Ai ≡ Bi for i ≤ q. If neither A < B nor B < A, we say that A and B are of the
same order and we denote A ≡ B. If F ⊆ R, then the set of all chains of F has a minimal
element, which is called a characteristic set of F and it is denoted by CS(F). A pr-pol f
is reduced with respect to a chain if it is reduced to every pr-pol in the chain. If f 6= 0 is
reduced with respect to CS(F), then CS(F ∪ {f }) < CS(F).
Lemma 2.2 A ⊂ P is a characteristic set of P if and only if there is no nonzero pr-pols in
P which are reduced with respect to A.
A difference ideal is a subset I of R = K{x1 , . . . , xn }, which is an algebraic ideal in R
and is closed under transforming. A difference ideal I is called reflexive if ηf ∈ I implies
f ∈ I for all η ∈ Tσ . Let S be a set of elements of R. The difference ideal generated by
S is denoted by [S]. Obviously, [S] is the set of all linear combinations of the pr-pols in S
and their transforms. The (ordinary or algebraic) ideal generated by S is denoted as (S). A
difference ideal I of R is called perfect if the presence in I of a product of transforms of an
element f of R implies f ∈ I. The perfect difference ideal generated by S is denoted as {S}.
A perfect ideal is always reflexive. A difference ideal I is called a prime difference ideal if it
is prime as an algebraic ideal.
Let A be a chain and IA the set of products of the initials of the pr-pols in A and their
transforms. The saturation ideal of A is defined as follows
sat(A) = [A] : IA = {f ∈ K{X} | ∃J ∈ IA , s.t.Jf ∈ [A]}.
Let B be an algebraic triangular set and IB the set of products of the initials of the pols
in B. The algebraic saturation ideal of B is defined as follows
a-sat(B) = (B) : IB = {f ∈ K[X] | ∃J ∈ IB , s.t.Jf ∈ (B)}.
Lemma 2.3 Let f, g ∈ R = K{X}. Then we can find non-negative integers sj and pr-pols
fi , r such that
I1s1 · · · Iksk g = f1 η1 f + · · · fl ηl f + r
where Ij is the transform of init(f ) such that uIj < ug and r is reduced with respect to f .
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X.S. Gao and G.L.Zhang
Proof: Let Σ = {ηuf |η ∈ Tσ , deg(g, ηuf ) ≥ deg(g, uf )}. Σ = ∅ if and only if g is reduced with
respect to f . Let η1 uf be the highest element of Σ with respect to the order. According to
the algebraic pseudo remainder formula, (η1 If )a1 g = g1 η1 f + r1 . r1 is reduced with respect
to η1 f about η1 uf . Let Σ1 = {ηur1 |η ∈ Tσ , deg(g, ηur1 ) ≥ deg(g, ur1 )}. Then every element
in Σ1 is strictly less than the highest element in Σ. Continue this process for Σ1 until we get
∅. Rearranging the symbols used above, we have I1s1 · · · Iksk g = f1 η1 f + · · · fl ηl f + r.
The pr-pol r in Lemma 2.3 is defined as the partial difference pseudo remainder of g with
respect to f , denoted as r = rprem(g, f ).
3. Coherent chains
For any chain A, after
following form:




A=



a proper renaming of the variables, we could write it as the
A1,1 (u, y1 ), . . . , A1,k1 (u, y1 )
A2,1 (u, y1 , y2 ), . . . , A2,k2 (u, y1 , y2 )
...
Ap,1 (u, y1 , . . . , yp ), . . . , Ap,kp (u, y1 , . . . , yp )
(1)
where lvar(Ai,j ) = yi and U = {u1 , . . . , uq } such that p + q = n. For c = 1, . . . , p, let
Ac = {Ac,1 (u, y1 , . . . , yc ), . . . , Ac,kc (u, y1 , . . . , yc )}.
Let hi ∈ Zm
∗ = {(z1 , . . . , zm )|zj is nonnegative integer}. We use A(h1 ,...,hl ) to denote the
following sequence of pr-pols in increasing ordering:
For s = l to 1 and for all h ≤ max{hs , vec(Ai,j , ys )| Ai,j ∈ ∪li=s Ai }, let Πh = {As,j | As,j ∈
As , h − vec(As,j ) º 0}, and As,j ∈ Πh the one with the least degree about its leader (if there
are more than one, choose the highest element w.r.t <). Then for all h̄ such that h − h̄ º 0
S
and h̃ = h̄ − vec(As,j ) º 0, let As = As {σ h̃ As,j }. Finally,
A(h1 ,...,hl ) = ∪li=1 Ai
It is easy to see that under the given variable order, A(h1 ,...,hl ) is an algebraic triangular set.
For a chain A and a pr-pol f , we introduce the following notations:
Af = A(vec(f,y1 ),··· ,vec(f,yp ))
(2)
We define rprem(f, A) = prem(f, Af ) where the variables and their transforms in f and Af
are treated as independent algebraic variables.
Lemma 3.1 Let f, A be as the above. There is a J ∈ IA such that uJ < uf , Jf ≡ r mod [A],
and r is reduced with respect to A.
Definition 3.2 Let A = A1 , . . . , Al be a chain in K{X} and vi = vec(Ai , uAi ), i = 1, . . . , l.
For any 1 ≤ i < j ≤ m, if class(Ai ) = class(Aj ) = t, let the least common multiple
transform of uAi and uAj be ηi,j uAi = ηj,i uAj . Then let ∆i,j = ηi,j Ai , ∆j,i = ηj,i Aj . If
rprem(∆i,j , A) = 0 and rprem(∆j,i , A) = 0P
for all i, j, we call A a coherent chain.
Let A = A1 , . . . , Al be a chain. g = i,j gi,j ηi,j Ai is called canonical if ηi,j Ai in the
expression are distinct elements in A(h1 ,...,hp ) for some h1 , . . . , hp ∈ Zm
∗ . In other words,
g ∈ (A(h1 ,...,hp ) ).
CS Method for Difference Polynomial Systems
85
Lemma 3.3 Let A be a coherent chain of form (1), A ∈ A, and η ∈ Tσ . Then there is a
J ∈ IA such that uJ < uηA and JηA has a canonical representation.
Proof: Let c = class(A). The pr-pols in A with class c are Ac,1 , · · · , Ac,i−1 , Ac,i = A, · · · , Ac,kc .
First, if uηA is not the multiple transform of any one of uA1 , · · · , uAi−1 , uAi+1 · · · , uAc,kc , then
ηA ∈ AηA . Second, suppose that uηA is the multiple transform of uAc,k , but ηA ∈ AηA .
Otherwise, we will prove this by induction on the ordering of uηA . Let the least common
transform of uA and uAc,k be uηi A = uηk Ac,k , ∆i,k = ηi A, η̄ηi = η, so ηA = η̄∆i,k . Since A is
a coherent chain, rprem(∆i,k , A) = 0. We have
¯ i,k = g1 τ1 B1 + g2 τ2 B2 + · · ·
J∆
where Bj ∈ A, τj Bj ∈ A∆i,j , and uJ¯ < u∆i,k , degree(∆i,k , u∆i,k ) ≥ degree(τ1 B1 , uτ1 B1 ),
u∆i,k = uτ1 B1 > uτ2 B2 > · · · . Let η̄ act on the two sides of the above equation and we get
η̄ J¯ · η̄∆i,j = η̄g1 · η̄τ1 B1 + η̄g2 · η̄τ2 B2 + · · ·
We denote it by
J1 ηA = ḡ 1 · ρ1 B1 + ḡ 2 · ρ2 B2 + · · ·
¯ uJ < ηA, ρj = η̄j τj . If ρ1 B1 is not of the first two cases, we continue the
where J1 = η̄ J,
1
above process on ρ1 B1 until we get (after rearrange the symbols properly)
J2 ηA = f1 · θ1 C1 + f2 · θ2 C2 + · · ·
where Cj ∈ A θj ∈ Tσ uηA = uθ1 C1 > uθ2 C2 > · · · and θ1 C1 is of the first two cases, any
θ2 C2 , θ3 C3 , · · · satisfy the induction hypothesis. Then there is J ∈ IA such that JηA has a
canonical representation.
P
Lemma 3.4 If A = A1 , . . . , Al is a coherent chain, for any f = gi,j ηj Ai , there is a J ∈ IA
such that J · f has a canonical representation, and uJ < max{uηj Ai }.
Proof: This is a direct consequence of Lemma 3.3.
4. Regular chains
4.1. Invertibility of algebraic polynomials
We will introduce some notations and results about invertibility of algebraic polynomials
with respect to an algebraic ascending chain(autoreduced set). These results are from [1, 3].
Let A = A1 , . . . , Am be a nontrivial triangular set in K[x1 , . . . , xn ] over a field K of characteristic zero. Let yi be the leading variable of Ai , y = {y1 , . . . , yp } and u = {x1 , . . . , xn }\y.
u is called the parameter set of A. We can denote K[x1 , . . . , xn ] as K[u, y]. Ii is the initial
of Ai . For a triangular set A, let IA be the set of products of the initials of the polynomials
in A.
Definition 4.1 Let A = A1 , A2 , . . . , Am be a nontrivial triangular set in K[u, y] with u as the
parameter set, and f ∈ K[u, y]. f is said to be invertible with respect to A if (f, A1 , . . . , As ) ∩
K[u] 6= {0} where s =class(f ). A is called regular if the initials of Ai are invertible with
respect to A1 , . . . , Ai−1 .
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X.S. Gao and G.L.Zhang
Theorem 4.2 [1, 3] Let A be a triangular set. Then A is a characteristic set of (A) : IA
iff A is regular.
Lemma 4.3 [3] A finite product of polynomials which are invertible with respect to A is also
invertible with respect to A.
Lemma 4.4 [3] A polynomial g is not invertible with respect to a regular triangular set A
iff there is a nonzero f in K[u, y] such that f g ∈ (A) and g is reduced with respect to A.
Lemma 4.5 Let f, g be algebraic polynomials in K[x1 , . . . , xn ] with class(f ) = class(g) = n.
f is irreducible and resultant(f, g, xn ) = 0 then f | g.
Proof: It is a corollary of the lemma 7.2.3 in Mishra’s book [8].
Lemma 4.6 [11, 8] Let A be an irreducible triangular set with a generic point α. Then for
any polynomial g, the following facts are equivalent.
• g is invertible with respect to A.
• ĝ 6= 0, where ĝ is obtained by substituting α into g.
• g 6∈ (A) : IA .
4.2. Properties of regular chains
Let A be a chain of form (1), f a pr-pol. f is said to be partial difference invertible, (or
invertible) with respect to A if it is invertible with respect to Af when f and Af are treated
as algebraic polynomials.
Definition 4.7 Let A = A1 , . . . , Al be a chain and Ii = init(Ai ). A is said to be (difference)
regular if ηIj is invertible with respect to A for any η ∈ Tσ and 1 ≤ j ≤ l.
Lemma 4.8 If A is a regular chain of form (1), then A(h1 ,...,hp ) is a regular algebraic triangular set for any h1 , . . . , hp ∈ Zm
∗ .
Proof: If A is difference regular, then by Definition 4.7, all ηIj are invertible with respect to
A. The initials of the pr-pols in A(h1 ,...,hp ) are all of the form ηIj and they are of ordering
lower than the highest ordering of the pr-pols in A(h1 ,...,hp ) . Then for any A ∈ A(h1 ,...,hp ) ,
AA ⊂ A(h1 ,...,hp ) . Therefore, A(h1 ,...,hp ) is a regular algebraic triangular set.
Lemma 4.9 If a chain A of form (1) is the characteristic set of [A] : IA , then for any
h1 , . . . , hp ∈ Zm
∗ , A(h1 ,...,hp ) is a regular algebraic triangular set.
Proof. By Lemma 4.2, we need only to prove that B = A(h1 ,...,hp ) is the characteristic set of
(B) : IB . Let W be the set of all the ηyj such that ηyj is of lower or equal ordering than an
η̄yj occurring in B. Then B ⊂ K[W ]. If B is not the characteristic set of (B) : IB , then there
is g ∈ (B) : IB ∩ K[W ] which is reduced with respect to B and is not zero. g does not contain
ηyi which is of higher ordering than those in W . As a consequence, g is also reduced with
respect to A. Since g ∈ (B) : IB ⊂ A : IA and A is the characteristic set of [A] : IA , g must
be zero, a contradiction.
The following is a difference version of the Rosenfeld Lemma [10]. The condition in this
lemma is stronger than that used in the differential Rosenfeld Lemma. The conclusion is
also stronger.
CS Method for Difference Polynomial Systems
87
Lemma 4.10 Let A be a coherent and regular chain, and r a pr-pol reduced with respect to
A. If r ∈ [A] : IA , then r = 0.
Proof. Let A = A1 , A2 , . . . , Al . Since r ∈ [A] : IA , there is a J1 ∈ IA such that J1 · r ≡
0 mod [A]. By Lemma 4.3, J1 is difference invertible with respect to A, i.e. there is a pr-pol
J¯1 and a nonzero N ∈ K[V ] such that
J¯1 · J1 ≡ N mod [A]
where V is the set of parameters of AJ1 as an algebraic triangular set. Hence,
N r ≡ J¯1 · J1 · r ≡ 0 mod [A].
Or equivalently,
N ·r =
X
gi,j ηi,j Aj .
(3)
Since A is a coherent chain, by Lemma 3.4, there is a J2 ∈ IA such that J2 · N · r has a
canonical representation in [A], where uJ2 < max{uηi,j Aj } in (3). That is
J2 · N · r =
X
ḡi,j ρi,j Aj ,
(4)
ij
where, uρi,j Aj are pairwise different. If max{uρi,j Aj } in (4) is lower in ordering than
max{uηi,j Aj } in (3), we have already reduced the highest ordering of uηi,j Aj in (3). Othdb
+ Rb . Substituting udρab Ab by − ρρaaRIbb in
erwise, assume uρa Ab = max{uρi,j Aj } and Ab = Ib · uA
b
(4), the left side keeps unchanged since uJ2 < uρa Ab , N is free of uρa Ab and deg(r, uρa Ab ) <
deg(ρa Ab , uρa Ab ). In the right side, the ρa Ab becomes zero, i.e. the max{uρi,j Aj } decreases.
Clearing denominators of the substituted formula of (4), we obtain a new equation:
X
(ρa Ib )t · J2 · N · r =
fij τi,j Aj .
(5)
Note that in the right side of (5), the highest ordering of τi,j Aj is less than uρa Ab and (ρa Ib )t ·J2
is invertible with respect to A and after multiplying a polynomial which is invertible with
respect to A and can be represented as a linear combination of τi,j Aj all of which is strictly
lower than uρa Ab . Repeating the above process, we can obtain a nonzero N̄ , such that
N̄ · r = 0.
Then r = 0. By Lemma 2.2, A is the characteristic set of [A] : IA .
The following is the main result in this paper.
Theorem 4.11 A chain A is the characteristic set of [A] : IA iff A is coherent and difference
regular.
Proof: If A is coherent and difference regular, then by Lemma 4.10, any pr-pol in [A] : IA
which is difference reduced with respect to A is zero. So A is a characteristic set of [A] : IA .
Conversely, let A = A1 , A2 , . . . , Al be a characteristic set of the saturation ideal [A] : IA and
Ii = init(Ai ). For any 1 ≤ i < j ≤ l, let r = rprem(∆i,j , A) as in Definition 3.2. Then
r is in [A] : IA and is difference reduced with respect to A. Since A is the characteristic
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X.S. Gao and G.L.Zhang
set of [A] : IA , r = 0. Then A is coherent. To prove that A is regular, for any 0 ≤ i ≤ l,
η ∈ Tσ we need to prove that f = ηIi is invertible with respect to A. Assume this is not
true. By definition, f is not invertible with respect to Af when they are treated as algebraic
equations. By Lemma 4.9, Af is a regular algebraic ascending chain. By Lemma 4.4, there is
a g 6= 0 which is reduced with respect to Ag (and hence A) such that f ·g ∈ (Af ) ⊂ [A]. Since
f = ηIi ∈ IA , g ∈ [A] : IA and g is reduced with respect to A. Since A is the characteristic
set of [A] : IA , we have g = 0, a contradiction. Hence, f = ηIi is invertible with respect to
A and A is difference regular.
Theorem 4.12 If A is a coherent and regular difference regular chain of form (1),then
[
[A] : IA =
(A(h1 ,...,hp ) ) : IA(h1 ,...,hp ) .
hi ∈Zm
∗
Proof: It is easy to see that [A] : IA ⊃
S
hi ∈Zm
∗
(A(h1 ,...,hp ) ) : IA(h1 ,...,hp ) . Since A is coherent and
regular, A is the characteristic set of [A] : IA . Then for f ∈ [A]
S : IA , we have rprem(f, A) =
prem(f, Af ) = 0, that is, f ∈ (Af ) : IAf . Hence [A] : IA ⊂
(A(h1 ,...,hp ) ) : IA(h1 ,...,hp ) .
hi ∈Zm
∗
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