Resolvent Systems of Difference Polynomial Ideals

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1 Resolvent Systems of Difference Polynomial Ideals Xiao-Shan Gao and Chun-Ming Yuan Key Laboratory of Mathematics Mechanization Institute of Systems Science, AMSS Academia Sinica, Beijing , China ABSTRACT In this paper, a new theory of resolvent systems is developed for prime difference ideals and difference ideals defined by coherent and proper irreducible ascending chains. Algorithms to compute such resolvent systems are also given. As a consequence, we prove that any irreducible difference variety is birationally equivalent to an irreducible difference variety of codimension one. As a preparation to the resolvent theory, we also prove that the saturation ideal of a coherent and proper ascending chain is unmixed in the sense that all its prime components have the same dimension and order. Categories and Subject Descriptors I.1.2 [SYMBOLIC AND ALGEBRAIC MANIPULA- TION]: Algorithms Algebraic algorithms General Terms Algorithms, Theory Keywords Resolvent, difference ascending chain, difference polynomial, difference variety, unmixed decomposition. 1. INTRODUCTION A classic result in algebraic geometry states that any irreducible variety is birationally equivalent to an irreducible hypersurface. Or equivalently, any finitely generated algebraic extension field can be generated with a single element, called the primitive element of the extension field. Algorithms to construct such hypersurfaces or primitive elements were proposed based on the methods of resultant computation by Trager [17] and Loos [11], the Gröbner basis method by Gianni and Mora [9], Kobayashi et al [9], and Yokoyama et al [21], and the characteristic set method by Gao-Chou [5, 7] and Wang-Lin [18]. The idea is to introduce a linear transformation of variables and show that the new equation Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC 06,July 9 12, 2006, Genova,Italy. Copyright2006ACM /06/ $5.00. system can be transformed into the following special form using various elimination theories R(x 0), I 1(x 0)x 1 + U 1(x 0),..., I n(x 0)x n + U n(x 0) where R, I i, U i are univariate polynomials in x 0. In [15], Ritt proved similar results for ordinary differential polynomial equation systems by introducing the concept of resolvents for a prime ideal. Kolchin further gave generalizations of the resolvent to the partial differential case [13]. In [1], Cluzeau and Hubert extended the concept of resolvent to regular differential ideals and proposed an algorithm to compute it. In [10], Grigoriev used the resolvent to give a differential elimination algorithm of elementary complexity. Following the work of Ritt[16], Cohn established the Difference Algebra [2]. Recently, elimination theories for difference polynomial systems were studied by Mansfield and Szanto [14], van der Hoeven [19], and Gao-Luo [8]. The concept of resolvent for an irreducible difference variety was also introduced by Cohn [2, 3]. In difference case, the resolvent is not a single difference polynomial anymore. In general, it is an irreducible difference variety of codimension one, which may be called the resolvent variety. But, the difference resolvent theory is not as complete as in the algebraic and differential cases. First, when establishing the birational equivalence between an irreducible variety V and its resolvent variety W, the operations of inversion need to be used. More precisely, the rational map is from W to E t V where E is difference operator and t an integer. Second, no algorithms were given to compute the resolvent. In this paper, a more complete difference resolvent theory is proposed. We prove that for an irreducible difference variety V, there exists a resolvent variety which is birational equivalent to V. The improved result is possible, because we prove that an irreducible difference variety can be represented by a coherent and strong irreducible ascending chain [8]. Based on this fact, we develop a resolvent theory with better properties. We also give algorithms to compute the resolvents. Furthermore, for a coherent and proper irreducible ascending chain (definition in Section 3), we gave an algorithm to construct a series of resolvent systems. In [8], we give an algorithm to decompose the zero sets of a set of difference polynomials into the zero sets of difference varieties represented by coherent and proper irreducible ascending chains. Combining this result and the result in this paper, it is always possible to represent the zero set of a difference polynomial system by a series of resolvent varieties. In order to establish the resolvent theory, we also prove that the saturation ideal defined by a coherent and proper 101

2 irreducible ascending chain is unmixed in the sense that all its prime components have the same dimension and order. Comparing to the differential case, the theory and algorithm for difference resolvents are generally much more difficult. Based on the ascending chain representation of the reflexive prime ideals introduced by us, we can obtain the order of the first resolvent polynomial explicitly and hence give an effective algorithm to compute it. In the case of proper ascending chains, we introduce a method of combination to prove the existence of resolvents. Based on the implementation of a characteristic set method for difference polynomial systems introduced in [8], we implement the algorithms proposed in this paper. Examples are given to illustrate the algorithms. The most time consuming part of the computation process is the computation of the characteristic set. 2. DIFFERENCE POLYNOMIALS A difference field F is a field with a third unitary operation E satisfying: for any a, b F, E(a + b) = Ea + Eb, E(ab) = Ea Eb, and Ea = 0 if and only if a = 0. Here, E is called the transforming operator or simply a transform of F. If a F, Ea is called the transform of a. E n a = E(E n 1 a) is known as the n th transform. If E 1 a is defined for all a F, we say that F is inversive. Every difference field has an inversive closure [2]. As an example, let K be the set of rational functions in variable x defined on the complex plane. Let E be the map: Ef(x) = f(x + 1), f K. Then K is a difference field with transforming operator E. This is an inversive field. In this paper, K is assumed to be this difference field. Let X = {x 1,..., x n} be indeterminants. Then K{X} = K{x 1,..., x n} is called an n-fold difference polynomial (abbr. r-pol) ring over K. Any r-pol P in K{X} is an ordinary polynomial in variables E k x j(k = 0, 1, 2,..., j = 1,..., n). For convenience, we also denote E k x j by x j,k. Let P K{X}. The class of P, denoted by cls(p ), is the least p such that P K{x 1,..., x p}. If P K, we set cls(p ) = 0. The order of P w.r.t x i, denoted by ord(p, x i), is the largest j such that x i,j occurs in P. When x i,j does not occur in P, we set ord(p, x i) = 1. If cls(p ) = p and ord(p, x p) = q, we called x p the leading variable and x p,q the lead of P, denoted as lvar(p ) and lead(p ), respectively. The leading coefficient of P as a univariate polynomial in lead(p ) is called the initial of P, and is denoted as init(p ). An n-tuple over K is of the form a = (a 1,..., a n), where the a i are in some difference extension field of K. Let P K{X}. To substitute an n-tuple a into P means to replace x i,j occurring in P with E j a i. Let P be a set of r-pols in K{X}. An n-tuple over K is called a solution of the equation set P=0 if the result of substituting the n-tuple into each r- pol in P is zero. We use Zero(P) to denote the set of solutions of P = 0. For an r-pol P, we use Zero(P/P ) to denote the set of solutions of P = 0 which are not solutions of P = 0. For instance, let P = Ex 1 x 1 + Ex 1 x 1. Then x 1 = 1 x+c(x) is a solution of P = 0, where c(x) is any function satisfying c(x + 1) = c(x). A field K is called aperiodic if there does not exist an integer n such that for all a K, E n a = a. Lemma 2.1 ((p201 [2])). Let K be an aperiodic field and P K{X} a nonzero r-pol. Then we can find an n- tuple (α 1,..., α n) K n such that P (α 1,..., α n) 0. A difference ideal is a subset I of K{X}, which is an algebraic ideal in K{X} and is closed under the transform. A difference ideal I is called reflexive if for an r-pol P, EP I implies P I. Let P K{X}. The difference ideal generated by P is denoted by [P]. The (algebraic) ideal generated by P is denoted as (P). A difference ideal I is called perfect if the presence in I of a product of powers of transforms of an r-pol P implies P I. The perfect difference ideal generated by P is denoted as {P}. A perfect ideal is always reflexive. A difference ideal I is called a prime ideal if for r-pols P and Q, P Q I implies P I or Q I. Let I K{X} be a reflexive prime ideal. Then I has a generic zero α which has the following property: an r- pol P I if and only if P (α) = 0 [2]. For a reflexive prime difference ideal I K{X}, we define the dimension of I as the difference transcendental degree of a generic zero α = (α 1,..., α n) of I over K[2]. Let I be a difference ideal. Rename X = {x 1,..., x n} as two subsets: U = {u 1,..., u q} and Y = {y 1,..., y p} (p+q = n). U is called a parametric set of I if I K{U} = {0} and y i Y, I K{U, y i} = {0}. The dimension of a reflexive prime ideal is the number of its parameters. Let U be a parametric set of a reflexive prime ideal I. The order of I w.r.t U, denoted as ord U I, is max Ys I K{U}[Y s]={0} where Y s is a finite set of y i,j. Let (β 1,..., β q, γ 1,..., γ p) be a generic zero of I corresponding to U and Y. Then, the order of I w.r.t U is the algebraic transcendental degree of (γ 1,..., γ p) over the extension field K β 1,..., β q of K[2]. The effective order of I w.r.t U is the maximum number of y ij which is algebraic independent over K{U}, where K{U} is the inversive closure of K{U} [2], and we denote it as Eord U I. It is clear that Eord U I ord U I. Let U be a parametric set of a reflexive prime ideal I, and (β 1,..., β q, γ 1,..., γ p) a generic zero of I where β 1,..., β q are corresponding to U. U is defined as ld U (I) = lim inf k Then the limit degree of I w.r.t [K 0(φ k ) : K 0(φ k 1 )], where K 0 = K β 1,..., β q and φ s = {γ ij, 1 i p, 0 j s}. Let D be a difference field. A difference kernel R over D is an extension field, D(a, a 1,..., a r), r 1, of D, each a i ), and an extension τ of E to an isomorphism of D(a, a 1,..., a r 1) onto D(a 1,..., a r), such that τa i = a i+1, i = 0, 1,..., r 1. (a 0 = a.) r is called the length of the kernel. Let φ r a r be an denoting a vector (a (1) i,..., a (n) i algebraic transcendental basis of the elements in a r over D(a,..., a r 1). Then the limit degree of R w.r.t φ is defined as D(a,..., a r) : D(a,..., a r 1, φ r). 3. DIMENSION, ORDER, AND DEGREE OF A PROPER IRREDUCIBLE CHAIN Let P 1,P 2 be two r-pols and lead(p 1) = y p,q. P 2 is said to be reduced w.r.t P 1 if deg(p 2, y p,q+i) < deg(p 1, y p,q) for any nonnegative integer i. An r-pol P 1 has higher rank than an r-pol P 2, denoted as P 1 > rank P 2, if 1). cls(p 1) > cls(p 2), 2). c = cls(p 1) = cls(p 2) and ord(p 1, x c) > ord(p 2, x c), or 3). c = cls(p 1) = cls(p 2), o = ord(p 1, x c) = ord(p 2, x c) and deg(p 1, x c,o) > deg(p 2, x c,o). If P 1 > rank P 2 and P 2 > rank P 1 are not valid, P 1 and P 2 are said to have the same rank, denoted as P 1 = rank P 2. A finite sequence of nonzero r-pols A = A 1,..., A p is called an ascending chain or simply a chain, if p = 1 and 102

3 A 1 0 or cls(a 1) > 0, A i < rank A j and A j is reduced w.r.t A i for 1 i < j p. A chain A is called trivial if cls(a 1) = 0. For a chain (algebraic or difference), we denote by I A the product of the initials of A, by I A the set of products of initials of A and their transforms. Example 3.1. Let us consider P 1 = Ey 2 y 2 + 1, P 2 = E 2 y + Ey K{y}. Since P 1 < rank P 2, deg(p 2, Ey)) < deg(p 1, Ey)) and deg(p 2, E 2 y) < deg(p 1, Ey), P 2 is reduced w.r.t P 1. Hence, P 1, P 2 is a chain. The saturation ideal of a chain A is defined as follows sat(a) = [A] : I A= {P K{X} J I A, s.t. JP [A]}. If B is an algebraic chain, we define a-sat(b) = {P k, s.t. I k BP (B)}. A chain A = A 1,..., A p is said to be of higher rank than another chain B = B 1,..., B s, denoted as A > rank B, if one of the following conditions holds: (1) 0 < j min{p, s}, such that i < j, A i = rank B i and A j > rank B j, or (2) s > p and A i = rank B i for i p. We use A 1 rank A 2 to denote the relation of either A 1 < rank A 2 or A 1 = rank A 2. It is known that this is a Notherian total order, that is, any strictly decreasing sequence of chains must be finite. A characteristic set of any r-pol set P is a chain contained in P and has the lowest rank. An r-pol is said to be reduced w.r.t a chain if it is reduced to every r-pol in the chain. It is known that [16]. Lemma 3.2. A P is a characteristic set of P if and only if there is no nonzero r-pol in P which is reduced w.r.t A. We can define the pseudo-remainder of an r-pol P w.r.t a chain A: rprem(p, A) and prove [8]: Lemma 3.3. Let P be an r-pol, A a chain, and R = rprem(p, A). Then there is a J I A with lead(j) < rank lead(p ) s.t. JP R mod [A] and R is reduced w.r.t A. For any chain A, after a proper renaming of the variables, we could write it as the following form. 8 < A = : A 1,1(U, y 1),..., A 1,k1 (U, y 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. The orders of A i,j in y i are increasing and the degrees of A i,j in lead(a i,j) are decreasing. Let o (i,j) = ord(a i,j, y i). U is called the parametric set of A. We define the dimension of A as dim(a) = U, the degree of A as deg(a) = Q p deg(a i,k i, y i,o(i,ki ), the order of A ) as ord(a) = P p o (i,1), and the effective order of A as Eord(A) = P p (o (i,1) m i) where m i is the minimal integer such that y i,mi occurs in A i,1. Let h 1,..., h m (m p) be nonnegative integers. We use A (h1,...,h m) to denote the following sequence of r-pols A 1,1, EA 1,1,..., E o (1,2) o (1,1) 1 A 1,1, A 1,2,..., A 1,k1, EA 1,k1,..., Eĥ1 o (1,k1) A 1,k1,..., A m,1, EA m,1,..., E o (m,2) o (m,1) 1 A m,1, A m,2,..., A m,km, (1) EA m,km,..., Eĥm o (m,km) A m,km (2) where ĥi is defined as follows: ĥ m = max{h m, o (m,km)} +1, and for i = m 1,..., 1, o i =max{order of y i(x) appears in A i+1,1, EA i+1,1,..., Eĥm o (m,km)a m,km }, ĥ i = max{h i, o i, o (i,ki )} +1. It is obvious that A (h1,...,h m) is an algebraic triangular set with parameters: P(A) = {y i,j 1 i p, 0 j ord(a i,1, y i) 1}. (3) For a chain A and an r-pol P, let A = A (0,...,0) (4) A P = A (ord(p,y 1 ),...,ord(p,y p)) Let A = A 1,..., A p be an algebraic nontrivial triangular set in K[x 1,..., x n] over a field K. Let y i be the leading variable of A i, Y = {y 1,..., y p} and U the parametric set of A. A polynomial f is said to be invertible w.r.t A if either f K[U] or (f, A 1,..., A s) K[U] {0} where y s is the leading variable of f. An r-pol P is said to be invertible w.r.t A if it is invertible w.r.t A P when P and A P are treated as algebraic polynomials. An r-pol P is called effective in variable y i if y i,0 = y i(x) occurs in P. P is called effective if P is effective in lvar(p ). Let A = A 1,..., A m be a difference chain in K{X} and k i = ord(a i, lvar(a i)), i = 1,..., m. For any 1 i < j m, if cls(a i) = cls(a j) = t, let ij = prem(e k j k i A i, A j, y t,kj ) be the algebraic pseudo-remainder of E k j k i A i w.r.t A j in variable y t,kj ; otherwise, let ij = 0. If rprem( ij, A) = prem( ij, A ) = 0, we call A a coherent difference chain. Definition 3.4. A chain A of the form (1) is said to be proper irreducible if A is an algebraic irreducible triangular set; and For c = 1,..., p, A c,1 is effective and Âc,1 is irreducible in K(η c 1)[y c(x),..., y c(x + f c)], where f c = ord(a c,1, y c), B c = A K{U, y 1,..., y c} (B 0 = ), η c is a generic point for the algebraic irreducible chain B c, and Âc,1 is obtained by substituting ηc 1 into Ac,1. Proper irreducible chains have the following properties[8]. Lemma 3.5. Let A be a coherent and proper irreducible chain of the form (1). If P is invertible w.r.t A, then EP is invertible w.r.t A. Lemma 3.6. Let A be a coherent and proper irreducible chain. Then A is a characteristic set of sat(a). Lemma 3.7. Let A be a coherent and proper irreducible chain of the form (1), and f K{U, Y }, g K{U}[P(A)]\{0}. If gf sat(a), then f sat(a). Definition 3.8. A proper irreducible chain A of the form (1) is said to be strong irreducible if for any nonnegative integers h i, A (h1,...,h p) is an irreducible triangular set in algebraic case. Lemma 3.9. [8] If A is a coherent and strong irreducible chain, then sat(a) is a reflexive prime ideal. Theorem Let A be a coherent and proper irreducible chain of the form (1).We have: (1) U is a parametric set of ideal sat(a). That is, sat(a) K{U} = {0} and i, sat(a) K{U, y i} = {0}. 103

4 T (2) If {sat(a)} = r P i is an irredundant intersection of a set of prime difference ideals, then i, U is a parametric set for P i and thus dim(p i) = dim(a). (3) Eord U P i = ord U P i = ord(a) and P i is reflexive. Let P i = sat(a i) and A i a chain under the same variable order as A. Then A i is strong irreducible. (4) deg(a) = P r ld U(P i). Proof: By Lemma 3.6, A is a characteristic set of sat(a). As a consequence, we have sat(a) K{U} =, since every non-zero r-pol in sat(a) K{U} is reduced w.r.t to A and hence must be zero. If there exists an i, such that sat(a) K{U, y i} = {0}, let h = P(A) and C = A (0,...,0,h,0,...,0), where h is at the i-th place. Let Y and U be the set of all y i,j and u k,l occurring in C. By Lemma 3.5, all the initials of C are invertible w.r.t C. Then Zero(a-sat(C)) is an unmixed algebraic ideal of dimension dim(a) = h in K(U )[Y ] [6]. On the other hand, a-sat(c) K(U )[y i,0,..., y i,h ] sat(a) K(U )[y i,0,..., y i,h ] = {0}. From this, we have dim(a-sat(c)) h + 1, a contradiction. This proves (1). Let {sat(a)} = r T P i be an irredundant intersection of prime difference ideals. Since i, {sat(a)} P i, for each i we have P i K{U, y i} = {0}. Then we need only to prove P i K{U} = {0} for all i. Suppose P 1 K{U} = {0}. Then T g K{U} P i and g 0. Since {sat(a)} = r is P i an irredundant representation, there exists an f K{U, Y }, T such that f r P i, f {sat(a)}. We have gf {sat(a)}. i=2 So there exist non negative integers s 0, s 1,..., s r, such that Π r i=0(e i (gf)) s i = Π r i=0(e i (g)) s i Π r i=0(e i (f)) s i sat(a). By Lemma 3.7, we have Π r i=0(e i (f)) s i sat(a). Therefore, f {sat(a)}, which contradicts to the choice of f. So U is a transform independent set of P i. Hence dim(p i) = U. This proves (2). Let o i = ord(a i,1, y i). It is clear that P(A) = P p oi. Since A is proper irreducible, for any y ij, j o i, there is a nonzero P sat(a) such that P K{U}[P(A), y i,j]. Since sat(a) P i, ord U P i P p oi. If there exists an i, say i = 1, such that P i K{U}[P(A)] {0}, then there exists a g 0, g P 1 K{U}[P(A)]. Also, there exists an f K{U, Y }, such that f r T i=2 P i, f {sat(a)} and gf {sat(a)}. There exist non negative integers s 0,..., s l such that Π l i=0(e i (gf)) s i sat(a). Since g is invertible w.r.t A, by Lemma 3.5, E i g is also invertible w.r.t A. So, there exists g K{U, Y }, such that g Π l i=0(e i g) s i = M 0, M K{U}[P(A)]. Then M Π l i=0(e i f) s i sat(a). By Lemma 3.7, Π l i=0(e i f) s i sat(a), so f {sat(a)}, a contradiction. Therefore, U P(A) is algebraic independent in P i and hence ord U P i P(A). Then ord U P i = ord(a). Let P(A) (T ) = {E T i y ij y ij P(A)}, where T = (T 1,..., T p) is a set of non negative integers. Notice that {sat(a)} = r P i is an irredundant T representation and A is effective. In each P i, using the properties of the transcendental degree, we have P i T K{U}[P(A) (T ) ] = {0}. So Eord U P i P(A). Since Eord U P i ord U P i = P(A), we have Eord U P i = ord U P i. Then P i is reflexive. In order to prove A i is strong irreducible, we need only to prove that A i is effective. Let A i,j be the first r-pol in A i with lvar(a i,j) = y j. We need only to prove that A i,j is effective in y j. We already proved that ord(a i,j, y j) = o j. Let P j 1={y i,k 1 i j 1, 0 k o i 1}. If A i,j is not effective, we may obtain an r-pol Q K{U}[P j 1, y j,1,..., y j,oj ] P i. This is impossible, because from P i K{U}[P j 1, y j,0,..., y j,oj 1] = {0} and Eord U P i = ord U P i, we have P i K[U, P j 1, y j,1,..., y j,oj ] = {0}. We proved (3). We denote a generic zero of A as η = (α ij, β ij), where α ij, β ij correspond to u ij, y ij respectively. By the proof of Theorem 4.2 in [8], we know that there exists a difference kernel R of length one: E : K(a 0) K(a 1), and α = {α ij 1 i q, j 0} a 1 is a parametric set of R, deg U R = K(a 1) : K(a 0, α) = deg(a). By Lemma 5 in chapter 6 of [2], we know that R has a finite number of principal realizations, denoted as V i, 1 i m, and the generic zero of V i is denoted as η i. We define Spec(V i) = {f K{U, Y} f(η i) = 0}, so sat(a) Spec(V i) and Spec(V i) is a prime difference ideal. Since sat(a) = T r Pi Spec(Vi), there exists a j such that Spec(V i) P j. Since P j and Spec(V i) have the same dimension and order, Spec(V i) = P j. So for each i, there is an l i such that Spec(V i) = P li, and the generic zero for each P j is a principal realization of R, so it must be some η i. Hence, there is a one to one corresponding between V i and P j. By Lemma 5 in chapter 6 of [2], we have deg(a) = deg U R = P m ld UV i = P r ld U(P i). This proves (4). If two ideals I 1 and I 2 have the same parametric set U and ord U I 1 = ord U I 2 and Eord U I 1 = Eord U I 2, then we say that I 1 and I 2 are of the same type. As a consequence of Theorem 3.10, we have the following corollaries (proofs omitted). Corollary If A is coherent and proper irreducible, all the irredundant prime divisors of sat(a) are of the the same type with sat(a). In other words, sat(a) is unmixed in the sense that all its essential prime ideals have the same parametric set, the same dimension, and the same order. As a consequence, we can define dim(sat(a)) = dim(a) and ord U (sat(a)) = ord(a). It is proved in [8] that if A is coherent and strong irreducible, then sat(a) is a reflexive prime ideal with U as a set of parameters. Furthermore, we have Corollary If A is a coherent and strong irreducible chain, then sat(a) is a reflexive prime ideal satisfying dim(sat(a)) = dim(a), ord U (sat(a)) = Eord U (sat(a)) = ord(a), and ld U (sat(a)) = deg(a). Corollary Let A be a coherent and proper irreducible chain, and r\ {sat(a)} = sat(a i) a decomposition of {sat(a)} as irredundant prime difference ideals, where A i are strong irreducible chains under the same variable order as that of A. Then the first r-pol of A i is the same as the first r-pol of A. Corollary Let A be a coherent and proper irreducible chain of the form (1). If {sat(a)} has only one prime component, then A is strong irreducible and sat(a) is a prime ideal. 104

5 4. RESOLVENT SYSTEM FOR A REFLEXIVE PRIME IDEAL It is proved in [8] that for a reflexive prime difference ideal I, we can choose a proper order of variables such that under this variable order I = sat(a) and A is coherent and strong irreducible. So, we will start our discussion from a coherent and strong irreducible chain. 4.1 Resolvent systems Theorem 4.1. Let A be a coherent and strong irreducible chain of the form (1), K an aperiodic difference field or U = 0, and λ 1,..., λ p variables. There exists Q K{λ 1,..., λ p, U} such that if σ 1,..., σ p satisfy Q(σ 1,..., σ p, U) 0, then the characteristic set of sat(a, w P p σiyi) under the variable order U < w < y i is of the following form R, R 1,..., R s, I 1y 1,0 V 1,..., I py p,0 V p (5) where R, R i, I i, V i K{U, w}. Furthermore, R is effective in w, ord(r, w) = ord(a), ord(i i, w) < ord(a), and ord(v p, w) ord(a). Proof: Denote B = A, w P p λiyi. Then B is also coherent and strong irreducible and sat(b) is a reflexive prime ideal. Since U is a parametric set of sat(b), we could treat K 0 = K U, λ as the ground field where λ = {λ 1,..., λ p}. We denote by ord X(Z) the number of an algebraic transcendental basis of Z and their transformations over K 0 X. Let (α 1,..., α p, ω) be a generic zero of sat(b), and α = {α 1,..., α p}. Then Eord(sat(A)) = Eord(sat(B)) = Eord(ω) + Eord ω(α) ord(ω) + ord ω(α) = ord(ω, α) (6) = ord(sat(a)) = Eord(sat(A)). The last equation is due to Corollary As a consequence, Eord(ω) = ord(ω). Let T (w) sat(b) be an r-pol in U and w with the lowest rank. Since sat(b) is a prime ideal, T (w) can be chosen as an irreducible r-pol. Also, T (w) must be the first element of the characteristic set of sat(b) under the variable order U < λ i < w < y i. Since Eord(ω) = ord(ω), by Corollary 3.12, we have Eord(T, w) = ord(t, w) o A. In other words, T is effective in w. Differentiating T w.r.t λ i0 and substituting α i for y i, we have T T + y i,0 λ i0 w (y1,...,y p,w)=(α 1,...,α P p p, 0 λ iα i ) = 0 Since (α, ω) is a generic zero of sat(b) and S i(α, ω) = 0, S i = T T λ i0 + y i0 sat(b), 1 i p. Let H be the T resultant of and T w.r.t w o. We have ord(h, w) < o. Then there exist r-pols A, B such that AT + B T = H. Let Q i = BS i + AT y i,0 = Hy i,0 + B T λ i0 sat(b). Let P i = rprem(q i, T, w o) = I i0y i0 +V i0. Then I i0 0 and P i sat(b), where I 0, V i0 K{U, λ, w}, ord(i i0, w) < ord(a), and ord(v i0, w) ord(a). Since T is irreducible and has the lowest order in w, we have I i0 / sat(b). From P i sat(b), we have Eord w(y i) = ord w(y i) = 0. By (6), we have Eord(T, w) = ord(t, w) = ord(a). Since T, P i sat(b), the characteristic set of sat(b) under the variable order U < λ i < w < y i must be of the form: T, T 1,..., T s, P 1,..., P p where T i K{U, λ, w}. Let Q K{U, λ} be the product of all the coefficients of rprem(i i0, B) as a polynomial in y i,j. Since K U is aperiodic, there exist σ 1,..., σ p K{U} such that Q(σ 1,..., σ p) 0. Let B be the chain A, w P p σiyi. We have I 0y i0 + V i0 sat(b), where I 0, V i0 are obtained by replacing each λ i with σ i in Q, V i0 respectively. Since Q(σ 1,..., σ p) 0, we have I 0 sat(b). We denote I i = I 0, V i = V i0. Then P i = I iy i,0 + V i sat(b). From ord(i 0, w) < ord(a) and ord(v i0, w) ord(a), we have ord(i i, w) < ord(a) and ord(v i, w) ord(a). As a consequence, a characteristic set of sat(b) must be of the form (5). Repeat the process for proving equation (6), we can show that Eord(R, w) = ord(r, w) = ord(a). Hence R is effective in w. This proves the theorem. Now, we may extend the following well-known result in algebraic geometry to difference case. Corollary 4.2. Any irreducible difference variety V is birationally equivalent to an irreducible difference variety of codimension one if the ground field K is aperiodic or V is of positive dimension. Proof: Let I be the set of r-pols which vanish on V. Then I is a reflexive prime ideal. Construct a resolvent (5) for I as done in Theorem 4.1. Let W = Zero(sat(R, R 1,..., R s)). The rational maps are defined as follows: M 1 : V W ; (U, y 1,..., y p) (U, σ iy i) M 2 : W V ; (U, w) (U, V1(U, w) Vp(U, w),..., I 1(U, w) I p(u, w) ). M 2 can be defined on W Zero( Q i Ii). From the construction of σ i, it is easy to show that M 1 and M 2 are inverse to each other. Hence V and W are birationally equivalent. We hence call sat(r, R 1,..., R s) the resolvent ideal and R, R 1,..., R s the resolvent system of V or I = sat(a). Note that from the results about resolvents in [2], we cannot obtain Corollary 4.2. The reason is that in [2], it is only proved that I iy i,mi V i I for some non-negative integer m i. So the construction of M 2 is not valid. In our case, due to the introduction of characteristic set, we can show that I iy i V i I, which leads to the result proved in this section. 4.2 Algorithm to compute resolvent system We first give an algorithm to compute the first element of the resolvent system for sat(a). Algorithm 4.3. Input: a coherent and strong irreducible chain A of the form (1) and a variable set Λ = {λ 1,..., λ p}. Output: P a T K{U, Λ, w} which is an r-pol in sat(a, w p λiyi) with the lowest rank w.r.t the variable order U < λ i < w < y i. Step 1 Let o = ord(a), A o = A (o,...,o), and U, Y the sets of u ij and y ij occurring in A o respectively. Then A o is an irreducible algebraic triangular set in K[U, Y ][8]. Step 2 Let A o = A o, w 0 P p λi,0yi,0,..., wo P p λ i,oy i,o and Λ o = {λ i,j, i = 1,..., p; j = 0,..., o}. Then it is clear that A o is an irreducible algebraic triangular set in K[Λ o, U, Y ]. Step 3 Compute a characteristic set C of a-sat(a o) in the polynomial ring K[Λ o, U, Y ] under the variable order U < λ i,j < w 0 < w 1 <... < w o < y i,j with methods proposed in [4, 20] and output the first element in C. 105

6 Proof of the correctness of Algorithm 4.3. Let B be a characteristic set of a-sat(a o) under the given variable order. It is clear that U and Λ o are in the parametric set of a-sat(a o). By the definition of the order for a chain, A o has o = ord(a) parameters of the form y i,j. In other words, a-sat(a o) is of dimension o over the base field K(U, Λ o). Since w i, i = 0,..., o are o + 1 linear combinations of y i,j, they must satisfy an algebraic relation. Let T be such a relation and with the lowest rank. Then T a-sat(a o) sat(a, w P p λiyi). From Theorem 4.1, we know that the r-pol in sat(a, w P p λiyi) with the lowest rank under the variable order U < Λ < w < y i is of order o in w. Hence T must involve w o and is an r-pol in sat(a, w P p λiyi) with the lowest rank w.r.t the variable order U < λ i < w < y i. Algorithm 4.4. Input: a coherent and strong irreducible chain A of form (1). Output: σ i K{U} and a characteristic set of sat(a, w P p σiyi) which is of form (5). Step 1 Let λ i, i = 1,..., p be p variables. With Algorithm 4.3, we may compute a T K{U, λ 1,..., λ p, w} which is an r-pol in sat(a, w P p λiyi) with the lowest rank w.r.t the variable order U < λ i < w < y i. Step 2 Let S be the resultant of and T w.r.t w o. Find T r-pols A, B such that AT + B T = S. Step 3 For i = 1,..., p, let Q i = B( T λ i,0 AT y i,0 = Sy i,0 +B T λ i0 rprem(q i, T ). sat(a, w P p T + y i,0 ) + λiyi), Ri = Step 4 Let B = A, w P p λiyi and Q K{U, λ} be the product of the coefficients of rprem(init(r i), B) as a polynomial in y i,j. By Lemma 2.1, we may select σ i K{U} s.t. Q(σ 1,..., σ p) 0. Step 5 Let P i = (R i) (λ1,...,λ p)=(σ 1,...,σ p). From the proof of Theorem 4.1, we know that P i = I iy i0 V i sat(a, w P p σiyi), Ii sat(a, w P p σiyi), and ord(i i, w) < ord(a). Step 6 Using the zero decomposition theorem proposed in [8], we may find the following decomposition under the variable order U < w < y 1 <... < y p Zero(sat(A, w σ iy i, P 1,..., P p)/i A) = Zero(A {w σ iy i, P 1,..., P p}/i A) = izero(sat(b i)/i A) where B i are coherent and proper irreducible after a proper renaming of the variables. Step P 7 By Theorem 4.1, a characteristic set of sat(a, w p σiyi, P1,..., Pp) is of the form (5). By Corollary 3.11, one of B i, say B 1, must have U as its parametric P set and ord(b 1) = ord(a). Then sat(a {w p σiyi}) must be the only prime component of {sat(b 1)}. By Corollary 3.14, B 1 is strong irreducible and sat(a {w P p σiyi}) = sat(b1). Output B1. In the above algorithm, we need to introduce p new parameters, which will increase the computational costs. In the following, we will give a probabilistic algorithm. Theorem 4.5. Let A be a coherent and strong irreducible chain of the form (1) and σ i, i = 1,..., p elements in K{U}. Using the zero decomposition theorem proposed in [8], we may find the following decomposition under the variable order U < w < y 1 <... < y p Zero(sat(A, w = Zero(A {w σ iy i)/i A) (7) σ iy i}/i A) = izero(sat(b i)/i A) where B i are coherent and proper irreducible chains after a proper renaming of the variables. If one of B i, say B 1, is of the form (5), then B 1 is strong irreducible and is a resolvent system for sat(a). The proof of Theorem 4.5 is omitted. Based on this theorem, we may just select p elements σ i from K{U} randomly and compute the zero decomposition (7). The probability of success is nearly one since by Theorem 4.1, the bad σ i are solutions of an r-pol equation Q = 0. Example 4.6. Consider the coherent and strong irreducible chain A = {y x, y 2 2,1 + y , y 2,2 y 2}. Let w = y 1 + y 2. Using the zero decomposition theorem proposed in [8], we find the following decomposition under the variable order w < y 1 < y 2: where B 1 is: Zero(sat(A, w y 1 y 2)) = Zero(A {w y 1 y 2}) = Zero(sat(B 1)) R = w (8 + 4w2 )w (16w2 + 6w 4 + 8x )w (4w6 + 8w x 2 48w 2 x 2 64w 2 x)w 2 1 +w8 + 8w 4 x x 4, R 1 = (2w 2 1 w + 2w3 4wx)w ( 2w2 1 w2 + w 4 1 3w4 + 4w x 2 )w 2 +4w 3 + w 5 8wx ww xw3 4wx 2, P 1 = ( 4w 3 4w 2 1 w + 8wx)y 1 + 4w w2 1 w2 + w 4 + w 4 1 8w 2 x + 4x 2, P 2 = ( 8wx+4w 2 1 w+4w3 )y 2 3w 4 2w 2 1 w2 +4w 2 1 +w4 1 +4x2. By Theorem 4.5, {R, R 1} is a resolvent system for sat(a). 5. RESOLVENT SYSTEM OF A PROPER IRREDUCIBLE CHAIN Generally, we do not know how to decide wether a chain is strong irreducible or not. On the other hand, we can decide whether a chain is proper irreducible. In this section, we will give an algorithm for the resolvent system of a proper irreducible chain. Theorem 5.1. Let A be a coherent and proper irreducible chain of the form (1) and λ 1,..., λ p difference variables. We assume that K is an aperiodic difference field or U = 0. Then there exists Q K{λ 1,..., λ p, U} such that if σ 1,..., σ p satisfy Q(σ 1,..., σ p, U) 0, the characteristic set of the perfect ideal {sat(a, w P p σiyi)} under the variable order U < w < y i is of the following form R, R 1,..., R s, I 1y 1,0 V 1,..., I py p,0 V p (8) where R, R i, I i, V i K{U, w}. Furthermore, R is effective in w and ord(r, w) = ord(a). Proof: Let o = ord(a), Y h = (y 1,0, y 1,1,..., y 1,h,..., y p,0,..., y p,h ), W h = (w 0,..., w h ), and Λ = {λ 1,..., λ p}. For any r- pol set P, we denote P = {P, w P p λiyi}. Let {sat(a)} = 106

7 rt P s be an irredundant decomposition of {sat(a)} as the s=1 intersection of prime difference ideals. Then {sat(a), w P p λiyi} = T r P i, where P s = {P s, w P λ iy i}. By Theorem 4.1, for each s, the characteristic set for P s under the variable order U < λ i < w < y i is of the following form: T s, T s,1,..., T s,ls, I s,1y s,0 V s,1,..., I s,py s,0 V s,p where T i, T i,j, I i,j, V i,j K{U, w} and ord(t s, w) = o, ord(i i,j) < o, and ord(v i,j) o. Let Ps = P s T P K{U, Λ}[Wo, Y 0]. Then P = {sat(a), w p λiyi} T K{U, λ}[w o, Y 0] = T r s=1 P s. Also, T s, I s,jy s,0 V s,j Ps. It is easy to see that Ps = a-sat(t s, I s,1y s,0 V s,1,..., I s,py s,0 V s,p) is an algebraic prime ideal. From the proof of Theorem 4.1, we know that I s,iy s,i V s,i are constructed from T s. Thus if T s = T t, Ps = Pt. Let Pi 0 = Pi 1 =... = Pi t (i 0 = 1) be the distinct Pi. We have T ik T ij for k j. So P = T t j=0 P i j is an irredundant representation and S k = Q k l=0 Ti T k l j=0 P i j. We will show that there exist I i, V i K{U, Λ, w}, such that I iy i0 V i P and ord(i i, w) < o. We will construct such r-pols by doing induction on k. Suppose that there exists J i,k y i,0 + W i,k k j=0pi j and ord(j i,k, w) < o. Since S k k j=0pi j and Q k = T ik+1 Pi k+1 are distinct r-pols of the same order w.r.t w and Q k is irreducible, the resultant H of S k and Q k w.r.t w o is not zero. From the property of the resultant, there exist A(w), B(w) K{U}[W o] such that H = AS k + BQ k. Let Then, we have J i,k+1 = BQ k J i,k + AS k I i,k+1, W i,k+1 = BQ k W i,k + AS k V i,k+1. J i,k+1 y i,0 W i,k+1 = AS k I i,k+1 y i,0 AS k V i,k+1 = H(I i,k+1 y i,0 V i,k+1 ) = 0 mod P i k+1 J i,k+1 y i,0 W i,k+1 = BQ k J i,k y i,0 BQ k W i,k = H(J i,k y i,0 W i,k ) = 0 mod k j=0 P i j Therefore, J i,k+1 y i,0 W i,k+1 k+1 j=0 P i j. Since J i,k+1 = HI i,k+1 0 mod Pi k+1 and J i,k+1 = HJ i,k 0 mod k j=0 Pi j, the resultant H of J i,k+1 and S k+1 w.r.t w o is not zero. We have r-pols A, B such that H = A J i,k+1 + B S k+1. Let P i = A (J i,k+1 y i,0 W i,k+1 ) + B S k+1 y i,0 = H y i,0 A W i,k+1 k+1 j=0 P i j. Then there exists P i = I iy i,0 V i P {sat(a), w P λ iy i}. Similar to the proof of Theorem 4.5, we may select an r-pol Q and σ i K{U}, such that when replacing λ i by σ i we have I i 0 and I iy i0 + V i {sat(a), w P σ iy i} {P i, w P σ iy i}. From the Theorem 4.1, the characteristic set of {P i, w P σ iy i} under variable order U < w < y i is of the following form S i, S i,1,..., S i,li, I i,1y i,0 V i,1,..., I s,py s,0 V s,p where S i, S i,j, I i,j, V i,j K{U, w} and ord(t s, w) = o. Let R be the product of the distinct S i. Then R i{p i, w P σiy i} = {sat(a), w P σ iy i} and must be such an r-pol with lowest rank. This proves the theorem. For a proper irreducible chain, we can find a resolvent system for sat(a). Theorem 5.2. Let A be a coherent and proper irreducible chain of the form (1), K an aperiodic difference field or U = 0. Then we can find σ 1,..., σ p K{U} such that Zero(sat(A, w X σ iy i)) = t Zero(sat(R i)) or {sat(a, w X σ iy i)} = t {sat(r i)} where R i is coherent and proper irreducible under the variable order U < w < y i and of the following form: R i, R i,1,..., R i,si, I i,1y 1,0 V i,1,..., I i,py p,0 V i,p (9) R i, R i,j, I i,j, V i,j K{U, w}. Furthermore, R i is effective in w and ord(r i, w) = ord(a). We call {R i, R i,1,..., R i,si }, i = 1,..., t, resolvent systems for sat(a). We will prove the theorem by giving an algorithm to compute the resolvent systems. Algorithm 5.3. Input: a coherent and proper irreducible chain A of the form (1). Output: the resolvent systems for sat(a). Step 1 Let B = A, w P p λiyi. Then B is also a coherent and proper irreducible chain. By Lemma 3.3, we have Zero(sat(B)/I B) = Zero(B/I B). Step 2 Using the difference zero decomposition theorem proposed in [8], under the variable order U < λ i < w < y 1 <... < y p, we may find a decomposition Zero(B/I B) = s Zero(sat(A i)/i B) (10) where A i are coherent and proper irreducible after a proper renaming of the variables. Step 3 Let T i, and P i,j be the r-pols with lowest rank in A i such that lvar(t i) = w, lvar(p i,j) = y j. By Theorem 4.1, the characteristic sets for all the prime components of sat(b) are of the form (5). By Corollary 3.11, {sat(a i)} is an unmixed ideal, and hence only for those A i with U as the parametric set and satisfying ord(t i, w) = ord(a) and ord(p i,j, y j) = 0, Zero(sat(A i)/j) is not redundant. We may simply assume that all the A i in (10) satisfy this condition. Step 4 For each A i, let S i,j = T i T λ j0 + y i j. By Corollary 3.13, the first r-pol in the characteristic set for each prime component of {sat(a i)} is also T i. S i,j is in each prime component of {sat(a i)} and hence S i,j {sat(a i)}. Let R i be the resultant of T i w o and A, B r-pols such that R i = A T i and T i w.r.t + BT i. Let Q i,j = AS i,j + BT iy j,0 = R iy i,0 + A T i λ j0 and P i,j = rprem(q i,j, T i). We have P i,j {sat(a i)}. Step 5 Let P i,j = I i,jy j,0 V i,j where I i,j, V i,j K{U, λ, w}. Since ord(i i,j, w) < ord(a) and ord(t i, w) = ord(a), I i,j is not in each of sat(a i). Hence I i,j is invertible w.r.t B. Let Q 0 be obtained by taking the successive resultant of I i,j and the r-pols in B Ii,j. Then Q 0 is not zero and in K{U}[P(A)], where P(A) is defined in (3). Let Q K{U, λ} be the product of the coefficients of Q 0 as a polynomial in y i,j. Select σ i K{U} such that Q(σ 1,..., σ p) 0. Step 6 For an r-pol P and a chain C, let P and C be obtained by replacing λ i with σ i. It is clear that P i,j {sat(a i)}. Since Q(σ 1,..., σ p) 0, Q 0 0. Then I i,j is invertible w.r.t B and hence not in sat(b). Due to (10), I i,j / {sat(a i)}. 107

8 Step 7 Using the difference zero decomposition theorem proposed in [8], we may find the following decomposition under the variable order U < w < y 1 <... < y p Zero(sat(A i) {P i,1,..., P i,p)/i Ai ) = Zero(A i {P i,1,..., P i,p}/i Ai ) = jzero(sat(b i,j)/i Ai ) where B i,j are coherent and proper irreducible chains. Step 8 Using the similar argument in Step 3, we only select those B i,j which are of the form (8). This is possible since P i,j are linear in y i,0. Output B i,j. We have proved the Theorem 5.2. In the following, we will give a probabilistic algorithm. Theorem 5.4. Let A be a coherent and proper irreducible chain of the form (1) and σ i, i = 1,..., p, elements in K{U}. Suppose that we find the following decomposition under the variable order U < w < y 1 <... < y p Zero(sat(A, w X σ iy i)) = t Zero(sat(R i)) where R i is proper irreducible and of the following form R i, R i,1,..., R i,si, I i,1y 1,0 V i,1,..., I i,py p,0 V i,p (11) R i, R i,j, I i,j, V i,j K{U, w} and ord(r i, w) = ord(a). Then {R i, R i,1,..., R i,si } are the resolvent systems for sat(a). The Proof is omitted. Example 5.5. Consider the chain A = {y x, y 2 2,1 + y }. Let w = y 1 + y 2. Using the zero decomposition theorem proposed in [8], we find the following decomposition under the variable order w < y 1 < y 2: Zero(sat(A, w y 1 y 2)) = Zero(A {w y 1 y 2}) = Zero(sat(B 1)) Zero(sat(B 2)) where B 2 = {R, R 1, P 1, P 2}, B 1, R, P 1, P 2 are given in Example 4.6 and R 1 = ( 2w1w 2 2w 3 + 4wx)w2 2 + ( 2w1w w1 4 3w 4 + 4w x 2 )w 2. By Theorem 5.4, {R, R 1} and {R, R 1} are the resolvent systems for sat(a). 6. CONCLUSION Intuitively speaking, resolvents can be used to establish a birational correspondence between the solutions of a set of equations and the solutions of equations in one variable. For difference equations, the theory of resolvent is not complete in several aspects. In this paper, we give a more complete theory of resolvents. For an irreducible difference variety V, we can construct a coherent and strong irreducible chain R in one variable such that V and Zero(sat(R)) are birationally equivalent. For a coherent and proper irreducible chain A, we can construct coherent and proper irreducible chains R i in one variable such that Zero(sat(A)) and izero(sat(r i)) are birationally equivalent. An interesting problem is to see whether izero(sat(r i)) in the proper irreducible case can be combined into one chain. That is, can we find a chain B such that Zero(sat(B)) = izero(sat(r i))? 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