On Partial Difference Coherent and Regular Ascending Chains

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 {σ 1,, σ m } a finite set of transforming operators over K. The order of an element η = σ o 1 1 σom m of T σ = {σ o 1 1 σom m j = 1,, m, o j 0} is ord(η) = Σ m j=1 o j. η is proper if ord(η) 0. The vector of an element η T σ is vec(η) = (i 1,, i m ) if η = σ i 1 1 σm im. For η 1, η 2 T σ, η 1 is a (proper) multiple transform of η 2 or η 2 is a (proper) factor transform of η 1 if η T σ such that η 1 = ηη 2 (ord(η) 0). It is denoted by vec(η 1 ) vec(η 2 ) 0( 0). Let X = {x 1,, x n } be a finite set of difference indeterminates over K and T σ X = {ηx i η T σ, i = 1,, n}. K{X} = K{x 1,, x n } = K[T σ X] denotes the partial difference ring of partial difference polynomials (pr-pol) in the indeterminates T σ X with coefficients in K. Let < be a tall ordering over T σ X defined as follows: η, θ T σ, 1 i, j m, ηx i > θx j 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 T σ X(w.r.t <) that appears in f, and we denote it by u f. We write f = I d u d f + + I 0. I d = init(f) is the initial of f. Let f K{X} and u f = ηx i. Then i and x i are called the class and leading variable of f, denoted as class(f) and lvar(f) respectively. We define vec(ηx i ) = vec(η) and vec(f, x j ) = vec(η), if ηx j = max{τx j 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 u f < u g or (u f = u g ) = 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, ηu g ) < deg(g, u g ), η T σ.

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{x 1,, x n } 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 = {ηx i ηx i = u A, 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 ρ t x i for some s,, t. Without loss of generality, we may assume that there is an infinite subset Λ 1 Λ every element of which is a proper multiple transform of ηx i. Continue the above process for Λ 1, we obtain η 1. Then are infinite sequences η, η 1, η 2,. Suppose η j x i = u Aj, we have deg(a 1, u A1 ) > deg(a 2, u A2 ) >, a contradiction. If A = {A 1,, A p } is a chain with A 1 < < A p, then we denote A as a sequence of pr-pols A = A 1,, A p. If A = A 1,, A p and B = B 1,, B q are two chains, we say that A < B if either there is some j min(p, q) such that A i B i for i < j and A j < B j, or q < p and A i B i 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 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{x 1,..., x n }, 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 T σ. 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 I A 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] : I A = {f K{X} J I A, s.t.jf [A]}. Let B be an algebraic triangular set and I B 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) : I B = {f K[X] J I B, s.t.jf (B)}. Lemma 2.3 Let f, g R = K{X}. Then we can find non-negative integers s j and pr-pols f i, r such that I s 1 1 Is k k g = f 1η 1 f + f l η l f + r where I j is the transform of init(f) such that u Ij < u g and r is reduced with respect to f.

84 X.S. Gao and G.L.Zhang Proof: Let Σ = {ηu f η T σ, deg(g, ηu f ) deg(g, u f )}. Σ = if and only if g is reduced with respect to f. Let η 1 u f be the highest element of Σ with respect to the order. According to the algebraic pseudo remainder formula, (η 1 I f ) a 1 g = g 1 η 1 f + r 1. r 1 is reduced with respect to η 1 f about η 1 u f. Let Σ 1 = {ηu r1 η T σ, deg(g, ηu r1 ) deg(g, u r1 )}. Then every element in Σ 1 is strictly less than the highest element in Σ. Continue this process for Σ 1 until we get. Rearranging the symbols used above, we have I s 1 1 Is k k g = f 1η 1 f + f l η l 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 a proper renaming of the variables, we could write it as the following form: A 1,1 (u, y 1 ),..., A 1,k1 (u, y 1 ) A A = 2,1 (u, y 1, y 2 ),..., A 2,k2 (u, y 1, y 2 ) (1)... A p,1 (u, y 1,..., y p ),..., A p,kp (u, y 1,..., y p ) where lvar(a i,j ) = y i and U = {u 1,..., u q } such that p + q = n. For c = 1,..., p, let A c = {A c,1 (u, y 1,..., y c ),..., A c,kc (u, y 1,..., y c )}. Let h i Z m = {(z 1,..., z m ) z j is nonnegative integer}. We use A (h1,...,h l ) to denote the following sequence of pr-pols in increasing ordering: For s = l to 1 and for all h max{h s, vec(a i,j, y s ) A i,j l i=s A i}, let Π h = {A s,j A s,j A s, h vec(a s,j ) 0}, and A s,j Π h 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 h such that h h 0 and h = h vec(a s,j ) 0, let A s = A s {σ ha s,j }. Finally, A (h1,...,h l ) = l i=1a i It is easy to see that under the given variable order, A (h1,...,h l ) is an algebraic triangular set. For a chain A and a pr-pol f, we introduce the following notations: A f = A (vec(f,y 1 ),,vec(f,y p)) (2) We define rprem(f, A) = prem(f, A f ) where the variables and their transforms in f and A f are treated as independent algebraic variables. Lemma 3.1 Let f, A be as the above. There is a J I A such that u J < u f, Jf r mod [A], and r is reduced with respect to A. Definition 3.2 Let A = A 1,..., A l be a chain in K{X} and v i = vec(a i, u Ai ), i = 1,..., l. For any 1 i < j m, if class(a i ) = class(a j ) = t, let the least common multiple transform of u Ai and u Aj be η i,j u Ai = η j,i u Aj. Then let i,j = η i,j A i, j,i = η j,i A j. If rprem( i,j, A) = 0 and rprem( j,i, A) = 0 for all i, j, we call A a coherent chain. Let A = A 1,..., A l be a chain. g = i,j g i,jη i,j A i is called canonical if η i,j A i in the expression are distinct elements in A (h1,...,h p) for some h 1,..., h p Z m. In other words, g (A (h1,...,h p)).

CS Method for Difference Polynomial Systems 85 Lemma 3.3 Let A be a coherent chain of form (1), A A, and η T σ. Then there is a J I A such that u J < u ηa and JηA has a canonical representation. Proof: Let c = class(a). The pr-pols in A with class c are A c,1,, A c,i 1, A c,i = A,, A c,kc. First, if u ηa is not the multiple transform of any one of u A1,, u Ai 1, u Ai+1, u Ac,kc, then ηa A ηa. Second, suppose that u ηa is the multiple transform of u Ac,k, but ηa A ηa. Otherwise, we will prove this by induction on the ordering of u ηa. Let the least common transform of u A and u Ac,k be u ηi A = u ηk A c,k, i,k = η i A, ηη i = η, so ηa = η i,k. Since A is a coherent chain, rprem( i,k, A) = 0. We have J i,k = g 1 τ 1 B 1 + g 2 τ 2 B 2 + where B j A, τ j B j A i,j, and u J < u i,k, degree( i,k, u i,k ) degree(τ 1 B 1, u τ1 B 1 ), u i,k = u τ1 B 1 > u τ2 B 2 >. Let η act on the two sides of the above equation and we get We denote it by η J η i,j = ηg 1 ητ 1 B 1 + ηg 2 ητ 2 B 2 + J 1 ηa = ḡ 1 ρ 1 B 1 + ḡ 2 ρ 2 B 2 + where J 1 = η J, u J1 < ηa, ρ j = η j τ j. If ρ 1 B 1 is not of the first two cases, we continue the above process on ρ 1 B 1 until we get (after rearrange the symbols properly) J 2 ηa = f 1 θ 1 C 1 + f 2 θ 2 C 2 + where C j A θ j T σ u ηa = u θ1 C 1 > u θ2 C 2 > and θ 1 C 1 is of the first two cases, any θ 2 C 2, θ 3 C 3, satisfy the induction hypothesis. Then there is J I A such that JηA has a canonical representation. Lemma 3.4 If A = A 1,..., A l is a coherent chain, for any f = g i,j η j A i, there is a J I A such that J f has a canonical representation, and u J < max{u ηj A i }. 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 = A 1,..., A m be a nontrivial triangular set in K[x 1,..., x n ] over a field K of characteristic zero. Let y i be the leading variable of A i, y = {y 1,..., y p } and u = {x 1,..., x n }\y. u is called the parameter set of A. We can denote K[x 1,..., x n ] as K[u, y]. I i is the initial of A i. For a triangular set A, let I A be the set of products of the initials of the polynomials in A. Definition 4.1 Let A = A 1, A 2,..., A m 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, A 1,..., A s ) K[u] {0} where s =class(f). A is called regular if the initials of A i are invertible with respect to A 1,..., A i 1.

86 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) : I A 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 fg (A) and g is reduced with respect to A. Lemma 4.5 Let f, g be algebraic polynomials in K[x 1,..., x n ] with class(f) = class(g) = n. f is irreducible and resultant(f, g, x n ) = 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 0, where ĝ is obtained by substituting α into g. g (A) : I A. 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 A f when f and A f are treated as algebraic polynomials. Definition 4.7 Let A = A 1,..., A l be a chain and I i = init(a i ). A is said to be (difference) regular if ηi j is invertible with respect to A for any η T σ and 1 j l. Lemma 4.8 If A is a regular chain of form (1), then A (h1,...,h p) is a regular algebraic triangular set for any h 1,..., h p Z m. Proof: If A is difference regular, then by Definition 4.7, all ηi j are invertible with respect to A. The initials of the pr-pols in A (h1,...,h p) are all of the form ηi j and they are of ordering lower than the highest ordering of the pr-pols in A (h1,...,h p). Then for any A A (h1,...,h p), A A A (h1,...,h p). Therefore, A (h1,...,h p) is a regular algebraic triangular set. Lemma 4.9 If a chain A of form (1) is the characteristic set of [A] : I A, then for any h 1,..., h p Z m, A (h1,...,h p) is a regular algebraic triangular set. Proof. By Lemma 4.2, we need only to prove that B = A (h1,...,h p) is the characteristic set of (B) : I B. Let W be the set of all the ηy j such that ηy j is of lower or equal ordering than an ηy j occurring in B. Then B K[W ]. If B is not the characteristic set of (B) : I B, then there is g (B) : I B K[W ] which is reduced with respect to B and is not zero. g does not contain ηy i which is of higher ordering than those in W. As a consequence, g is also reduced with respect to A. Since g (B) : I B A : I A and A is the characteristic set of [A] : I A, 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] : I A, then r = 0. Proof. Let A = A 1, A 2,..., A l. Since r [A] : I A, there is a J 1 I A such that J 1 r 0 mod [A]. By Lemma 4.3, J 1 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 J 1 N mod [A] where V is the set of parameters of A J1 as an algebraic triangular set. Hence, Nr J 1 J 1 r 0 mod [A]. Or equivalently, N r = g i,j η i,j A j. (3) Since A is a coherent chain, by Lemma 3.4, there is a J 2 I A such that J 2 N r has a canonical representation in [A], where u J2 < max{u ηi,j A j } in (3). That is J 2 N r = ij ḡ i,j ρ i,j A j, (4) where, uρ i,j A j are pairwise different. If max{u ρi,j A j } in (4) is lower in ordering than max{u ηi,j A j } in (3), we have already reduced the highest ordering of u ηi,j A j in (3). Otherwise, assume u ρaab = max{u ρi,j A j } and A b = I b u d b A b + R b. Substituting u d b ρ aa b by ρar b ρ ai b in (4), the left side keeps unchanged since u J2 < u ρaab, N is free of uρ a A b and deg(r, u ρaab ) < deg(ρ a A b, u ρaab ). In the right side, the ρ a A b becomes zero, i.e. the max{u ρi,j A j } decreases. Clearing denominators of the substituted formula of (4), we obtain a new equation: (ρ a I b ) t J 2 N r = f ij τ i,j A j. (5) Note that in the right side of (5), the highest ordering of τ i,j A j is less than u ρaab and (ρ a I b ) t J 2 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 τ i,j A j all of which is strictly lower than u ρaab. Repeating the above process, we can obtain a nonzero N, such that N r = 0. Then r = 0. By Lemma 2.2, A is the characteristic set of [A] : I A. The following is the main result in this paper. Theorem 4.11 A chain A is the characteristic set of [A] : I A 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] : I A which is difference reduced with respect to A is zero. So A is a characteristic set of [A] : I A. Conversely, let A = A 1, A 2,..., A l be a characteristic set of the saturation ideal [A] : I A and I i = init(a i ). For any 1 i < j l, let r = rprem( i,j, A) as in Definition 3.2. Then r is in [A] : I A and is difference reduced with respect to A. Since A is the characteristic

88 X.S. Gao and G.L.Zhang set of [A] : I A, r = 0. Then A is coherent. To prove that A is regular, for any 0 i l, η T σ we need to prove that f = ηi i is invertible with respect to A. Assume this is not true. By definition, f is not invertible with respect to A f when they are treated as algebraic equations. By Lemma 4.9, A f is a regular algebraic ascending chain. By Lemma 4.4, there is a g 0 which is reduced with respect to A g (and hence A) such that f g (A f ) [A]. Since f = ηi i I A, g [A] : I A and g is reduced with respect to A. Since A is the characteristic set of [A] : I A, we have g = 0, a contradiction. Hence, f = ηi i 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] : I A = Proof: It is easy to see that [A] : I A h i Z m h i Z m (A (h1,...,h p)) : I A(h1,...,hp). (A (h1,...,h p)) : I A(h1. Since A is coherent and,...,hp) regular, A is the characteristic set of [A] : I A. Then for f [A] : I A, we have rprem(f, A) = prem(f, A f ) = 0, that is, f (A f ) : I Af. Hence [A] : I A (A (h1,...,h p)) : I. A(h1,...,hp) References h i Z m [1] P. Aubry, D. Lazard, and M.M. Maza, On the Theory of Triangular Sets, Journal of Symbolic Computation, 28, 105-124, 1999. [2] I. Bentsen, The Existence of Solutions of Abstract Partial Difference Polynomial, Trans. of AMS, 158, 373-397, 1971. [3] D. Bouziane, A. Kandri Rody, H. Maârouf, Unmixed-dimensional Decomposition of a Finitely Generated Perfect Differential Ideal, Journal of Symbolic Computation, 31, 631-649, 2001. [4] R.M. Cohn, Difference Algebra, Interscience Publishers, 1965. [5] R.M. Cohn, Manifolds of Difference Polynomials, Trans. of AMS, 64, 133-172, 1948. [6] X.S. Gao and Y. Luo, A Characteristic Set Method for Difference Polynomial Systems, International Conference on Polynomial System Solving, Nov. 24-26, 2004. Submitted to JSC. [7] M.V. Kondratieva, A.B.Levin, A.V.Mikhalev and E.V.Pankratiev, Differential and Difference Dimension Polynomials, Kluwer Academic Publishers, 1999. [8] B. Mishra, Algorithmic Algebra, Springer-Verleg, New York, 1993. [9] J.F. Ritt and J.L. Doob, Systems of Algebraic Difference Equations, American Journal of Mathematics, 55, 505-514, 1933. [10] A. Rosenfeld, Specialization in Differential Algebra, Trans. Am. Math. Soc., 90, 394-407, 1959. [11] W.T. Wu, Basic Principle of Mechanical Theorem Proving in Geometries, (in Chinese) Science Press, Beijing, 1984; Springer, Wien, 1994.