Criteria for Finite Difference Groebner Bases of Normal Binomial Difference Ideals
aa r X i v : . [ c s . S C ] J a n Criteria for Finite Difference Gröbner Bases ofNormal Binomial Difference Ideals ∗ Yu-Ao Chen and Xiao-Shan GaoKLMM, UCAS, Academy of Mathematics and Systems ScienceThe Chinese Academy of Sciences, Beijing 100190, China
Abstract
In this paper, we give decision criteria for normal binomial difference polynomial ideals in the univariatedifference polynomial ring F { y } to have finite difference Gröbner bases and an algorithm to computethe finite difference Gröbner bases if these criteria are satisfied. The novelty of these criteria lies in thefact that complicated properties about difference polynomial ideals are reduced to elementary properties ofunivariate polynomials in Z [ x ] . Keywords.
Difference algebra, binomial difference ideal, Gröbner basis, difference Gröbner basis.
Difference algebra founded by Ritt and Cohn aims to study algebraic difference equations in a similar way thatpolynomial equations are studied in commutative algebra and algebraic geometry [5, 14, 18, 21]. The Gröbnerbasis invented by Buchberger is a powerful tool for solving many mathematical problems [4]. The conceptsof difference Gröbner bases was extended to linear difference polynomial ideals in [11, 14, 15] and nonlineardifference polynomial ideals in [11]. Many applications of difference Gröbner bases were given [9, 14–16].Since difference polynomial ideals can be infinitely generated, their difference Gröbner bases are generallyinfinite. Even for finitely generated difference polynomial ideals, their difference Gröbner bases could beinfinite as shown by Example 2.2 in this paper. This makes it impossible to compute difference Gröbner basesfor general difference polynomial ideals and thus it is a crucial issue to give criteria for difference polynomialideals to have finite difference Gröbner bases.Let F be a difference field and y a difference indeterminate. In this paper, we will give decision criteriafor normal binomial difference polynomial ideals in F { y } to have finite difference Gröbner bases and analgorithm to compute these finite difference Gröbner bases under these criteria. A difference ideal I in F { y } is called normal if MP ∈ I implies P ∈ I for any difference monomial M in F { y } and P ∈ F { y } . I iscalled binomial if it is generated by difference polynomials with at most two terms [6, 7].For f ∈ Z [ x ] , let f + , f − ∈ N [ x ] be the positive part and the negative part of f such that f = f + − f − . For h = (cid:229) mi = a i x i ∈ N [ x ] , denote y h = (cid:213) mi = ( s i y ) a i , where s is the difference operator of F . Then any differencemonomial in F { y } can be written as y g for some g ∈ N [ x ] . For a given f ∈ Z [ x ] with a positive leadingcoefficient, we consider the following binomial difference polynomial ideal in F { y } : I f = sat ( y f + − y f − ) = [ { y h + − y h − | h = g f , g ∈ Z [ x ] } ] ∗ Partially supported by a grant from NSFC 11101411. F , { h ∈ Z [ x ] | lt ( h ) = h + } . F , { h ∈ Z [ x ] | hg ∈ F for some monic polynomial g ∈ Z [ x ] } . We prove that I f has a finite difference Gröbner basis if and only if f ∈ F . This criterion is then extended togeneral normal binomial difference ideals in F { y } .The decision of f ∈ F is quite nontrivial and we give the following criteria for f ∈ F based on the rootsof f :1. if f has no positive roots, then f ∈ F ;2. if f has more than one positive roots (with multiplicity counted), then f F ;3. if f has one positive root x + and a root z such that | z | > x + , then f F ;4. if f has one positive root x + and a root z such that | z | = x + , then we can compute another f ∗ ∈ Z [ x ] and x ∗ ∈ R > such that f ∗ ( x ∗ ) = f ∗ ( w ) = | w | = x ∗ imply w = x ∗ , and f ∗ ( w ) = | w | 6 = x ∗ imply | w | < x ∗ . Furthermore, f ∈ F if and only if f ∗ ∈ F ;5. if f / ∈ F has a unique positive real root x + and x + <
1, then f F ;6. if f ( ) = z of f satisfies | z | <
1, then f ∈ F if and only if f ( x ) / ( x − ) ∈ Z [ x d ] forsome d ∈ N > and f ( x )( x d − ) / ( x − ) ∈ F .With these criteria, only one case is open: f has a unique positive real root x + , x + >
1, and x + > | z | for anyother root z of f . We conjecture that f ∈ F in the above case based on numerical computations. If I f has afinite difference Gröbner basis according to one of the six criteria listed above, we also give an algorithm tocompute it.As far as we know the above criteria are the first non-trivial ones for a difference polynomial ideal to havea finite difference Gröbner basis. The novelty of these criteria lies in the fact that complicated properties aboutdifference polynomial ideals are reduced to elementary properties of univariate polynomials in Z [ x ] .The rest of this paper is organized as follows. In Section 2, preliminaries on Gröbner basis for differencepolynomial ideals are given. In Section 3, criteria for normal binomial difference ideals in F { y } to have finitedifference Gröbner bases are given. In Section 4, criteria for f ∈ F and an algorithm to compute the finitedifference Gröbner basis of I f under these criteria are given. In Section 5, we propose an approach based oninteger programming to find g such that f g ∈ F and give a lower bound for deg ( g ) in certain cases. An ordinary difference field, or simply a s -field, is a field F with a third unitary operation s satisfying: forany a , b ∈ F , s ( a + b ) = s ( a ) + s ( b ) , s ( ab ) = s ( a ) s ( b ) , and s ( a ) = a =
0. We call s the difference or transforming operator of F . A typical example of s -field is Q ( l ) with s ( f ( l )) = f ( l + ) . Inthis paper, we use s - as the abbreviation for difference or transformally.For a in any s -extension ring of F and n ∈ N > , s n ( a ) is called the n -th transform of a and denoted by a x n ,with the usual assumption a = x =
1. More generally, for p = (cid:229) si = c i x i ∈ N [ x ] , denote a p = (cid:213) si = ( s i a ) c i . For instance, a x + x + = ( s ( a )) s ( a ) a . It is easy to check that a p satisfies the properties of powers [7].2et S be a subset of a s -field G which contains F . We will denote Q ( S ) = { s k a | k ∈ N , a ∈ S } , F { S } = F [ Q ( S )] . Now suppose Y = { y , . . . , y n } is a set of s -indeterminates over F . The elements of F { Y } arecalled s -polynomials over F in Y . A s -polynomial ideal I , or simply a s -ideal, in F { Y } is a possiblyinfinitely generated ordinary algebraic ideal satisfying s ( I ) ⊂ I . If S is a subset of F { Y } , we use ( S ) and [ S ] to denote the algebraic ideal and the s -ideal generated by S .A monomial order in F { Y } is called compatible with the s -structure, if y x k i < y x k j for k < k . Onlycompatible monomial orders are considered in this paper. When a monomial order is given, we use LM ( P ) and LC ( P ) to denote the largest monomial and its coefficient in P respectively, and LT ( P ) = LC ( P ) LM ( P ) the leading term of P . Definition 2.1. G ⊂ F { Y } is called a s -Gröbner basis of a s -ideal I if for any P ∈ I , there exist m ∈ N and G ∈ G such that ( LM ( G )) x m | LM ( P ) . From the definition, G is a s -Gröbner basis of I if and only if Q ( G ) is a Gröbner basis of I treatedas an algebraic polynomial ideal in F [ Q ( Y )] . Note that I is generally an infinitely generated ideal and theconcept of infinite Gröbner basis [12] is adopted here. From this observation, we may see that a s -Gröbnerbasis satisfies most of the properties of the usual algebraic Gröbner basis. For instance, G is a s -Gröbner basisof a s -ideal I if and only if for any P ∈ I , we have grem ( P , Q ( G )) =
0, where grem ( P , Q ( G )) is the normalform of P modulo Q ( G ) in the theory of Gröbner basis. The concepts of reduced s -Gröbner bases could besimilarly introduced. A s -polynomial Q is called s -reduced w.r.t. another s -polynomial P if there does notexist a k ∈ N such that LM ( P ) x k divides any monomial in Q . Then, a s -Gröbner G basis is called reduced, ifany P ∈ G is s -reduced w.r.t G \ { P } . It is easy to see that a s -ideal has a unique reduced s -Gr¨bner basis.The following example shows that even a finitely generated s -ideal may have an infinite s -Gröbner basis.As a consequence, there exist no general algorithms to compute the s -Gröbner basis. Example 2.2.
Let I = [ y y x − y x y , y y − ] . Assume y < y < y . Then under a compatible monomialorder, the reduced s -Gröbner basis of I ∩ F { y , y } is { y y x i − y x i y | i ∈ N > } . The elimination ranking R on Q ( Y ) = { s k y i | ≤ i ≤ n , k ∈ N } is used in this paper: s k y i > s l y j if and onlyif i > j or i = j and k > l , which is a total order over Q ( Y ) . By convention, 1 < s k y j for all k ∈ N .Let f be a s -polynomial in F { Y } . The greatest y x k j w.r.t. R which appears effectively in f is calledthe leader of f , denoted by ld ( f ) and correspondingly y j is called the leading variable of f , denoted bylvar ( f ) = y j . The leading coefficient of f as a univariate polynomial in ld ( f ) is called the initial of f and isdenoted by init f .Let p and q be two s -polynomials in F { Y } . q is said to be of higher rank than p if ld ( q ) > ld ( p ) orld ( q ) = ld ( p ) = y x k j and deg ( q , y x k j ) > deg ( p , y x k j ) . Suppose ld ( p ) = y x k j . q is said to be Ritt-reduced w.r.t. p ifdeg ( q , y x k + l j ) < deg ( p , y x k j ) for all l ∈ N .A finite sequence of nonzero s -polynomials A : A , . . . , A m is said to be a difference ascending chain , orsimply a s -chain , if m = A = m > A j > A i and A j is Ritt-reduced w.r.t. A i for 1 ≤ i < j ≤ m . A s -chain A can be written as the following form [8] A : A , . . . , A k , . . . , A p , . . . , A pk p (1)where lvar ( A i j ) = y c i for j = , . . . , k i , ord ( A i j , y c i ) < ord ( A il , y c i ) and deg ( A i j , ld ( A i j )) > deg ( A il , ld ( A il )) for3 < l . The following are two s -chains A : y x − , y y − , y x − A : y − , y x − y , y − , y x + y (2)Let A : A , A , . . . , A t be a s -chain with I i as the initial of A i , and P any s -polynomial. Then there existsan algorithm, which reduces P w.r.t. A to a s -polynomial R that is Ritt-reduced w.r.t. A and satisfies therelation t (cid:213) i = I e i i · P ≡ R , mod [ A ] , (3)where the e i ∈ N [ x ] and R = prem ( P , A ) is called the s -Ritt-remainder of P w.r.t. A [8].A s -chain C contained in a s -polynomial set S is said to be a characteristic set of S , if S doesnot contain any nonzero element Ritt-reduced w.r.t. C . Any s -polynomial set has a characteristic set. Acharacteristic set C of a s -ideal J reduces to zero all elements of J .Let A : A , . . . , A t be a s -chain, I i = init ( A i ) , y x oi l i = ld ( A i ) . A is called regular if for any j ∈ N , I x j i is invertible w.r.t A [8] in the sense that [ A , . . . , A i − , I x j i ] contains a nonzero s -polynomial involving no y x oi + k l i , k = , , . . . . To introduce the concept of coherent s -chain, we need to define the D -polynomial first. If A i and A j have distinct leading variables, we define D ( A i , A j ) =
0. If A i and A j ( i < j ) have the same leadingvariable y l , ld ( A i ) = y x oi l , and ld ( A j ) = y x oj l , then o i < o j [8]. Define D ( A i , A j ) = prem (( A i ) x oj − oi , A j ) . Then A is called coherent if prem ( D ( A i , A j ) , A ) = i < j [8]. Both A and A in (2) are regular and coherent s -chains.Let A be a s -chain. Denote I A to be the minimal multiplicative set containing the initials of elements of A and their transforms. The saturation ideal of A is defined to besat ( A ) = [ A ] : I A = { P ∈ F { Y } : ∃ m ∈ I A , mP ∈ [ A ] } . The following result is needed in this paper.
Theorem 2.3. [8, Theorem 3.3] A s -chain A is a characteristic set of sat ( A ) if and only if A is regular andcoherent. We also need the concept of algebraic saturation ideal. Let C be an algebraic triangular set in F [ x , . . . , x n ] and I the product of the initials of the polynomials in C . Then defineasat ( C ) = { P ∈ F [ x , . . . , x n ] | ∃ k ∈ N , I k P ∈ ( C ) } . s -Gröbner basis for a binomial s -ideal A s -monomial in Y can be written as Y f = (cid:213) ni = y f i i , where f = ( f , . . . , f n ) t ∈ N [ x ] n . A nonzero vector f =( f , . . . , f n ) t ∈ Z [ x ] n is said to be normal if the leading coefficient of f s is positive, where s is the largestsubscript such that f s =
0. For f ∈ Z [ x ] n , let f + , f − ∈ N n [ x ] denote respectively the positive part and the negativepart of f such that f = f + − f − . Then gcd ( Y f + , Y f − ) = f ∈ Z [ x ] n . If f ∈ Z [ x ] n is normal, then Y f + > Y f − and LT ( Y f + − c Y f − ) = Y f + under a monomial order compatible with the s -structure.A s -binomial in Y is a s -polynomial with at most two terms, that is, a Y a + b Y b where a , b ∈ F and a , b ∈ N [ x ] n . A s -ideal in F { Y } is called binomial if it is generated by, possibly infinitely many, s -binomials[7]. We have 4 roposition 2.4 ( [7]) . A s -ideal I is binomial if and only if the reduced s -Gröbner basis for I consists of s -binomials. Let m be the multiplicative set generated by y x j i for i = , . . . , n , j ∈ N . A s -ideal I is called normal iffor M ∈ m and P ∈ F { Y } , MP ∈ I implies P ∈ I . Normal s -ideals in F { Y } are closely related with the Z [ x ] -modules in Z [ x ] n [7, 13], which will be explained below. We first introduce a new concept. Definition 2.5. A partial character r on Z [ x ] n is a homomorphism from a Z [ x ] -module L r in Z [ x ] n to themultiplicative group F ∗ satisfying r ( x f ) = ( r ( f )) x = s ( r ( f )) for f ∈ L r . A Z [ x ] -module generated by h , . . . , h m ∈ Z [ x ] n is denoted as ( h , . . . , h m ) Z [ x ] . Let r be a partial characterover Z [ x ] n and f = { f , . . . , f s } a reduced Gröbner basis of the Z [ x ] -module L r = ( f ) Z [ x ] . For h ∈ Z [ x ] n and H ⊂ L r , denote P h = Y h + − r ( h ) Y h − and P H = { P h | h ∈ H } . Introduce the following notations associatedwith r : I + ( r ) : = [ P L r ] = [ Y f + − r ( f ) Y f − | f ∈ L r ] (4) A + ( r ) : = P f = { Y f + − r ( f ) Y f − , . . . , Y f + s − r ( f s ) Y f − s } . (5)It is shown that [7] A + ( r ) is a regular and coherent s -chain and hence is a characteristic set of sat ( A + ( r )) by Theorem 2.3. Furthermore, we have Theorem 2.6.
The following conditions are equivalent.1. I is a normal binomial s -ideal in F { Y } .2. I = I + ( r ) for a partial character r over Z [ x ] n .3. I = sat ( A + ( r )) for a partial character r over Z [ x ] n .Furthermore, for f ∈ Z [ x ] n , Y f + − c Y f − ∈ I ⇔ f ∈ L r and c = r ( f ) . As a direct consequence of Proposition 2.4 and Theorem 2.6, we have
Corollary 2.7.
Let r be a partial character over Z [ x ] n . Then P L r is a s -Gröbner basis of I + ( r ) . Note that for f ∈ Z [ x ] n , either f or − f is normal and we need only consider the normal vectors in the s -Gröbner basis. So, for simplicity, we may assume that all given vectors are normal. We have the followingcriterion for the s -Gröbner basis of normal binomial s -ideals. Corollary 2.8.
Let r be a partial character over Z [ x ] n and H ⊂ L r . Then P H is a s -Gröbner basis of I + ( r ) if and only if for any normal g ∈ L r , there exist h ∈ H and j ∈ N , such that g + − x j h + ∈ N [ x ] n .Proof: By Corollary 2.7, P L r is a s -Gröbner basis of I + ( r ) . Then P H is a s -Gröbner basis of I + ( r ) if andonly if for any normal g ∈ L r , there exist h ∈ H and j ∈ N such that LM ( x j P h ) | LM ( P g ) , which is equivalentto g + − x j h + ∈ N [ x ] n . Example 2.9.
Let f = [ − x , x − ] , L = ( f ) Z [ x ] , and r the trivial partial character on L, that is, r ( h ) = for h ∈ L. Then P f = y y x − y x y . By Theorem 2.6, I + ( r ) = sat ( P f ) . By Corollary 2.7, a s -Gröbner basisof I + ( r ) is { Y g + − Y g − | g = h f , h ∈ Z [ x ] , lc ( h ) > } . By Example 2.2, sat ( P f ) = [ P f , y y − ] ∩ Q { y , y } =[ y y x i − y x i y | i ∈ N > ] , and a reduced s -Gröbner basis of I + ( r ) is { y y x i − y x i y | i ∈ N > } . Criteria for finite s -Gröbner basis In this section, we will give a criterion for the s -Gröbner basis of a normal binomial s -ideal in F { y } to befinite, where y is a s -indeterminate. Without loss of generality, we assume r ( h ) = r over Z [ x ] and h ∈ L r . s -polynomial In this section, we consider the simplest case: n = L r = ( f ) Z [ x ] is generated by one polynomial f ∈ Z [ x ] .We will see that even this case is highly nontrivial. For g ∈ Z [ x ] , we use lc ( g ) , lm ( g ) , and lt ( g ) to represent theleading coefficient, leading monomial, and leading term of g , respectively.In the rest of this section, we assume f ∈ Z [ x ] and lc ( f ) >
0. Then P f = y f + − y f − and LT ( P f ) = y f + under a monomial order compatible with the s -structure. By Theorem 2.6, all normal binomial s -ideals in F { y } whose characteristic set consists of a single s -polynomial can be written as the following form: I f = sat ( P f ) = [ y h + − y h − | h = f g ∈ ( f ) Z [ x ] , ∀ ( g ∈ Z [ x ] , lc ( g ) > )] . (6)In this section, we will give a criterion for I f to have a finite s -Gröbner basis. Define F , { f ∈ Z [ x ] | lt ( f ) = f + } . F , { f ∈ Z [ x ] | f g ∈ F for some monic polynomial g ∈ Z [ x ] } . (7)We now give the main result of this section, which can be deduced from Lemma 3.3 and Lemma 3.7. Theorem 3.1. I f in (6) has a finite s -Gröbner basis under a monomial order compatible w.r.t the s -structureif and only if f ∈ F . For two polynomials h and h ∈ Z [ x ] , denote h (cid:23) h if h − h ∈ N [ x ] . For h and h ∈ N [ x ] , we have h (cid:23) h if and only if y h | y h . Lemma 3.2.
If f ∈ F , then { P f } is a s -Gröbner basis of I f .Proof: For g ∈ ( f ) Z [ x ] with lc ( g ) > ∃ h ∈ Z [ x ] with lc ( h ) > g = f h . Since f ∈ F , we havelt ( f ) = f + . Then, x deg ( h ) f + = lt ( h ) f + / lc ( h ) (cid:22) lt ( h ) f + = lt ( h ) lt ( f ) = lt ( g ) (cid:22) g + . By Corollary 2.8, { P f } is a s -Gröbner basis of I f . Lemma 3.3.
If f ∈ F , then I f has a finite s -Gröbner basis.Proof: Let h = f g ∈ F , where g is monic. Then lc ( h ) = lc ( f ) and lt ( h ) = lt ( f ) lm ( g ) = h + . I deg ( h ) = I f T F [ y , y x , · · · , y x deg ( h ) ] is a polynomial ideal in a polynomial ring with finitely many variables, which has afinite Gröbner basis denoted by G deg ( h ) . Let P u ∈ I f and lc ( u ) >
0. If deg ( u ) deg ( h ) , then there exists a P t ∈ G deg ( h ) such that t (cid:22) u . Otherwise, we have deg ( u ) > deg ( h ) and lc ( u ) ≥ lc ( f ) . Then x deg ( u ) − deg ( h ) h + = x deg ( u ) − deg ( f ) − deg ( g ) lt ( f ) lm ( g )= x deg ( u ) − deg ( f ) lt ( f ) = x deg ( u ) − deg ( f ) lc ( f ) lm ( f ) = lc ( f ) lm ( u ) (cid:22) lt ( u ) (cid:22) u + . Since that P h ∈ I deg ( h ) , by Corollary 2.8, G deg ( s ) is a finite s -Gröbner basis of I f .6 orollary 3.4. Let f ∈ F , h = g f ∈ F , g a monic polynomial in Z [ x ] , and D = deg ( h ) . Then the Gröbnerbasis of the polynomial ideal I D = I f T F [ y , y x , · · · , y x D ] is a finite s -Gröbner basis for I f . From the proof of Lemma 3.3, we have
Example 3.5. f = x + x + ∈ F , because ( x − ) f = x − ∈ F . The finite s -Gröbner basis is G = { y x + x + − , y x − y } . Let D be R or Z . We will use the following new notation D > [ x ] , { n (cid:229) i = a i x i | n ∈ N , ∀ i ( a i ∈ D > ) } . Lemma 3.6. N [ x ] ⊆ F .Proof: Let g = a n x n + a n − x n − + · · · + a ∈ N [ x ] with d = max { d ∈ N | x d | g } the multiplicity of f at 0.Then a d >
0. Let s = ( x n − d + x n − d − + · · · + ) g = a n x n − d + ( a n + a n − ) x n − d − + · · · + ( a n + · · · + a d ) x n +( a n − + · · · + a d ) x n − + · · · + a d x d . Rewrite s = b n − d x n − d + · · · + b d x d . Then s / x d ∈ Z > [ x ] . Let M = ⌈ max { b i − / b i | d + i n − d }⌉ +
1. Then ( x − M ) s = b n − d x n − d + + ( b n − d − − Mb n − d ) x n − d + · · · +( b d − Mb d + ) x d + − Mb d x d ∈ F . So both s and g are in F . Lemma 3.7.
If f F , then I f does not have a finite s -Gröbner basis.Proof: Suppose otherwise, I f has a finite s -Gröbner basis G = P H , where H = { f , · · · , f l } ⊂ Z [ x ] with eachlc ( f i ) >
0. Since f has the lowest degree in ( f ) Z [ x ] , we have f ∈ H .Let H c , { h ∈ H | lc ( h ) = lc ( f ) } . Since f / ∈ F , we have H c T F = /0. By Lemmas 3.2 and 3.6, for all h ∈ H c , h + has at least two terms and h − has at least one term. For u ∈ Z [ x ] with lc ( u ) >
0, define a function f deg ( u ) = deg ( u ) − ( deg ( u + − lt ( u ))) (8)which is the degree gap between the first two highest monomials of u + . Suppose h is an element in H c suchthat f deg ( h ) = max { f deg ( h ) | h ∈ H c } . h exists because f ∈ H c = /0 and H c is a finite set. Denote lt ( h ) , ax n ,˜ h , h − lt ( h ) , lt ( ˜ h + ) , bx m , and ˜˜ h + , ˜ h + − lt ( ˜ h + ) . Then h = ax n + bx m + ˜˜ h + − h − . Since h F , we have ab >
0. Let c , ⌈ b / a ⌉ ≥ s = ( x n − cx m ) h = ax n + x n ˜˜ h + + cx m h − − ( ac − b ) x m + n − cx m ˜ h + − x n h − . We have s + (cid:22) s , ax n + x n ˜˜ h + + cx m h − , and f deg ( s ) = deg ( s ) − deg ( s + − lt ( s )) ≥ f deg ( s ) = deg ( s ) − deg ( s + − lt ( s )) > n − m = f deg ( h ) = deg ( h ) − deg ( h + − lt ( h )) . Since P H is a s -Gröbner basis of I f , there exist h ∈ H and j ∈ N such that t = s + − x j h + ∈ N [ x ] . We claimlt ( t ) = lt ( s + ) . If h ∈ H c , then f deg ( s ) > f deg ( h ) . Note that deg ( s + ) = deg ( x j h ) implies that the coefficient of thesecond largest monomial of s + − x j h is negative contradicting to the fact s + − x j h ∈ N [ x ] . As a consequence, wemust have deg ( s + ) > deg ( x j h ) and the claim is proved in this case. Now let h ∈ H \ H c . Since lc ( h ) > lc ( s ) = lc ( f ) , we have deg ( x j h ) < deg ( s ) which implies lt ( t ) = lt ( s + ) . The claim is proved. The fact lt ( t ) = lt ( s + ) implies that when computing the normal form P u = grem ( P s , Q ( P H )) , we always have lt ( u ) = lt ( s ) . As aconsequence, P u = P H is a s -Gröbner basis of I f and s ∈ ( f ) Z [ x ] .Note that the proof of Lemma 3.7 gives a method to construct infinitely many elements in a s -Gr¨bner basisas shown in the following example. 7 xample 3.8. Let f = x − x + / ∈ F . In the proof of Lemma 3.7, c = ⌈ b / a ⌉ = and s = ( x − ) f = x + x − x − . Repeat the above procedure to s , we obtain s = ( x − x ) s = x + x + x − x − x .Then f deg ( f ) < f deg ( s ) < f deg ( s ) and P s i is in a s -Gröbner basis for all i. Thus any s -Gröbner basis of I f isinfinite. We can show that a minimal s -Gröbner basis is G = { y x i + − y x i | i ∈ Z > } S { y x i + + − y x i + + x i | i ∈ Z > } . s -Gröbner bases for normal binomial s -ideals In this section, we consider the general normal binomial s -ideals in F { y } . By Theorem 2.6, all normalbinomial s -ideals in F { y } can be written as the following form: I G = sat ( P G ) = [ y g + − y g − | ∀ g ∈ ( G ) Z [ x ] , lc ( g ) > ] (9)where G = { g , . . . , g t } ⊂ Z [ x ] (10)is a reduced Gröbner basis of the Z [ x ] -module L = ( G ) Z [ x ] . Gröbner bases in Z [ x ] have the following specialstructure [7]. Lemma 3.9.
Let G = { g , . . . , g k } be a reduced Gröbner basis of a Z [ x ] -module in Z [ x ] , g < · · · < g k , and lt ( g i ) = c i x d i ∈ N [ x ] . Then1) ≤ d < d < · · · < d k .2) c k | · · · | c | c and c i = c i + for ≤ i ≤ k − .3) c i c k | g i for ≤ i < k. If e b is the primitive part of g , then e b | g i for < i ≤ k. Here are two Gröbner bases in Z [ x ] : { , x } , { , x , x + } .In the rest of this section, let L = ( G ) Z [ x ] for G defined in (10) and define L i , { f ∈ L | lc ( f ) = c t = lc ( g t ) } (11) L t , { f ∈ L i | f has minimal degree in L i }} . (12) Theorem 3.10. I G has a finite s -Gröbner basis if and only if L i T F = /0 .Proof: Suppose L i T F = /0 and let g ∈ L i T F . Then I G T k [ y , y x , · · · , y x deg ( g ) ] has a finite Gröbner basisdenoted by G ≤ deg ( g ) . Let P u ∈ I G and lc ( u ) >
0. If deg ( u ) ≤ deg ( g ) , then there exists a P h ∈ G ≤ deg ( g ) suchthat h (cid:22) u . Otherwise, we have deg ( u ) > deg ( g ) and lc ( u ) ≥ lc ( g ) . Then x deg ( u ) − deg ( g ) g + = x deg ( u ) − deg ( g ) lt ( g ) = x deg ( u ) − deg ( g ) lc ( g ) lm ( g ) = lc ( g ) lm ( u ) (cid:22) lt ( u ) (cid:22) u + . By Corollary 2.8, G ≤ deg ( g ) is a finite s -Gröbner basis of I G , since P g is in G ≤ deg ( g ) .We will prove the other direction by contradiction. Suppose that L i ∩ F = /0 and I G has a finite s -Gröbner basis P H = { P u , · · · , P u k } . Let H = { u , · · · , u k } , and H c = H T L i . Since grem ( P g t , Q ( P H )) =
0, wehave H c = /0 and let u be an element of H c with maximal f deg which is defined in (8). Since L i ∩ F = /0, byLemma 3.6 u + contains at least two terms and u − =
0. Similar to the proof of Lemma 3.7, we can constructan s ∈ Z [ x ] ∩ L such that f deg ( s ) > f deg ( u ) and lc ( s ) = lc ( u ) . Then, grem ( P s , Q ( P H )) = P H is a s -Gröbner basis. 8 orollary 3.11. If I G has a finite s -Gröbner basis, then g ∈ F .Proof: Let e b be the primitive part of g . Then by Lemma 3.9, e b | h for any h ∈ L . By Theorem 3.10, e b andhence g is in F . Corollary 3.12.
If L t T F = /0 and in particular g t ∈ F , then I G has finite s -Gröbner Basis. The following example shows that g t ∈ F is not a necessary condition for the s -Gröbner basis to be finite. Example 3.13.
Let G = { ( x − ) , ( x − )( x + ) } . Then ( x − )( x + )( x − ) + ( x − ) = x − x − ∈ F ⊂ F , and hence I G has a finite s -Gröbner basis. On the other hand, we will show ( x − )( x + ) / ∈ F in Example 4.10. In order to give another criterion, we need the following effective Polya Theorem.
Lemma 3.14 ( [17]) . Suppose that f ( x ) = n (cid:229) j = a n x n ∈ R [ x ] is positive on [ , ¥ ) and F ( x , y ) the homogenizationof f . Then for N f > n ( n − ) L l − n, ( + x ) N f f ( x ) ∈ R > [ x ] , where l = min { F ( x , − x ) | x ∈ [ , ] } and L = max { k ! ( n − k ) ! n ! | a k |} . Corollary 3.15.
If there exists an h ∈ L with no positive real roots, then I G has a finite s -Gröbner basis.Proof: Write h = x m h such that h ( ) =
0. By Lemma 3.14, there exists an N ∈ N such that h = ( x + ) N h ∈ Z > [ x ] . Take a sufficiently large N such that deg ( h ) > d t = deg ( g t ) . Then there exists a sufficiently large M ∈ N , such that g = x m ( x deg ( h ) − deg ( g t )+ g t − Mh ) ∈ F . Since g ∈ L i , by Lemma 3.10, I has a finite s -Gröbner Basis. F and s -Gröbner basis computation In Section 3, we prove that sat ( P f ) has a finite s -Gröbner basis if and only if f ∈ F . In this section, we willgive criteria and an algorithm for f ∈ F . If f ∈ F , we also give an algorithm to compute the finite s -Gröbnerbasis.From the definition of F , a necessarily condition for f ∈ F is lc ( f ) >
0. Also, it is easy to show that f ∈ F if and only if cx m f ∈ F for positive integers c and m . So in the rest of this paper, we assume f = n (cid:229) i = a n x i ∈ Z [ x ] such that n >
0, lc ( f ) = a n > f ( ) = a =
0, and gcd ( a , a , . . . , a n ) = In this subsection, we will study whether f ∈ F by examining properties of the roots of f ( x ) = Lemma 4.1.
If f ∈ Z [ x ] has no positive real roots, then f ∈ F .Proof: By Lemma 3.14, there exists an N ∈ N , such that ( x + ) N f ∈ Z > [ x ] ⊆ N [ x ] . By Lemma 3.6, ( x + ) N f ∈ N [ x ] ⊆ F , and thus f ∈ F .By Lemma 4.1, we need only consider those polynomials which have positive roots.9 emma 4.2. Let f = a n x n + · · · + a ∈ F . Then f has a simple and unique positive real root x + , and for anyroot z of f , we have | z | ≤ x + .Proof: Since f ∈ F \ Z , the number of sign differences of f is one. Then by Descartes’ rule of signs [1],the number of positive real roots of f (with multiplicities counted) is one or less than one by an even number.Then f has a simple and unique positive real root x + . For any root z of f , since − a i ≥ i = , . . . , n − a n | z | n = | a n z n | = | − a n − z n − − · · · − a | ≤ − a n − | z | n − − · · · − a . (13)Thus f ( | z | ) f has at least one real root in [ | z | , ¥ ) . Since f has a unique positive real root x + , wehave | z | x + .We now consider those f which has a root z = x + and | z | = x + . Such a z must be either − x + or a complexroot. Lemma 4.3.
Let f = a n x n + · · · + a ∈ F and x + the unique positive root of f . If f has a root z = x + but | z | = x + , then we have1. z d f ∈ R > and z is a simple root of f , where d f = gcd { i | a i = } > .2. f is a polynomial in x d f : f = b f ◦ x d f , where ◦ is the function composition. Furthermore, b f ( w ) = and | w | = x d f + imply w = x d f + .3. f has exactly d f roots with absolute value x + : { z | f ( z ) = , | z | = x + } = { z k x + | z = e p i d f , k = , . . . , d f } ,where i = √− .Proof: Let z = x + be a root of f such that | z | = x + . Then f ( | z | ) = f ( x + ) = a n | z | n + a n − | z | n − + · · · + a = | − a n − z n − − · · · − a | = − a n − | z | n − − · · · − a . The above equation ispossible if and only if − a i z i ∈ R > for each i ≤ n − a i =
0. Also note, z n = ( − a n − | z | n − − · · · − a ) / a n ∈ R > . Then, z i ∈ R > for each i ≤ n and a i =
0. Note that z m ∈ R > and z k ∈ R > imply z m − k ∈ R > . As aconsequence, z d f ∈ R > for d f = gcd { i | a i = } . Since z = x + , we have d f >
1. Part 1 of the lemma is proved.From the definition of d f , f is a polynomial of x d f : f ( x ) = b f ( x ) ◦ ( x d f ) . It is easy to see that b f ( x ) ∈ F .Let b f ( x ) = b k x k + · · · + b x + b . Then gcd { j | b j = } =
1. By the first part of this lemma, we know x d f + is theonly root of f whose absolute value is x d f + . Since z d f and x d f + are both the unique positive real roots of b f ( x ) ,we have z d f = x d f + and hence z is a simple root of f . Part 2 of the lemma is proved. Part 3 of the lemma comesfrom the fact z d f = x d f + is the unique positive real root of f and f ( z ) = b f ( z d f ) = Corollary 4.4.
If f ∈ F has at least one positive real root x + , then x + is the unique positive real root of f ,x + is simple and for any root z of f , x + > | z | . If f has a root z = x + satisfying | z | = x + , then z is simple, andz d ∈ R > for some d ∈ N > , or equivalently, the argument of z satisfies Arg ( z ) / p ∈ Q . Example 4.5. f = ( x − )( x − x + ) / ∈ F , because the root z = + i satisfies | z | = √ but z d / ∈ R > forany d ∈ N . The following example shows that the multiplicity for a root z satisfying | z | < x + could be any number. Example 4.6.
For any n , k ∈ N > , ( x + ) n ( x − k ) ∈ F . Let n = , ( x + )( x − k ) ∈ F . Let f ( x ) = ( x + ) and f n + ( x ) = f n ( x )( x ⌊ deg ( f n ) / ⌋ + + ) for n > . Then we have ( x + ) n + | f n ( x ) , f n ( x ) ∈ Z > [ x ] , and allcoefficients of f n are either or . Thus, f n ( x )( x − k ) ∈ F and ( x + ) n ( x − k ) ∈ F by definition. emma 4.7. Let q ( x ) ∈ Z [ x ] be a primitive irreducible polynomial and d ∈ N > . Then ( q ) Z [ x ] T Z [ x d ] =( e q ( x d )) Z [ x d ] , where e q ∈ Z [ x ] is primitive and irreducible and e q ( x d ) m = R u ( u d − x d , q ( u )) for some m ∈ N . Weuse R u to denote the Sylvester resultant w.r.t. the variable u. Furthermore, the roots of e q ( x ) are { z d | q ( z ) = } .Proof: Let q ( x ) = a (cid:213) nj = ( x − z j ) , z d = e p i / d , and R ( x d ) = R u ( u d − x d , q ( u )) = d (cid:213) l = q ( z l d x ) . We claim that R ( x d ) is primitive. We have lc ( R u ( u d − x d , q ( u ))) = lc ( (cid:213) d l = q ( z l d x )) = a d . Let c ∈ Z be aprime factor of a d or a . Since q is primitive, q = ( mod c ) . Let q ( x ) = bx m + · · · ( mod c ) . Then lt ( R ( x d )) = lt ( (cid:213) d l = q ( z l d x )) = (cid:213) d l = b ( z l d x ) m = b d x d m = ( mod c ) . So c ∤ R ( x d ) and thus R ( x d ) is primitive.Since Q [ x d ] is a PID and R ( x d ) ∈ ( q ) Q [ x ] T Q [ x d ] , there exists a primitive polynomial e q ∈ Z [ x ] such that ( e q ( x d )) Q [ x d ] = ( q ) Q [ x ] T Q [ x d ] . Since q ( x ) | e q ( x d ) and q is irreducible, e q ( x ) must be irreducible. Since both q ( x ) and e q ( x ) are primitive, we can deduce ( e q ( x d )) Z [ x d ] = ( q ) Z [ x ] T Z [ x d ] from ( e q ( x d )) Q [ x d ] = ( q ) Q [ x ] T Q [ x d ] .Since q ( x ) | e q ( x d ) , Z d = { z k d z j | k = , . . . , d , j = , . . . , n } is a subset of the roots of e q ( x d ) . Let S ( x ) bethe square-free part of R ( x ) ∈ Z [ x ] , which is also primitive. Since Z d contains exactly the roots of R ( x d ) and S ( x d ) , we have S ( x ) | e q ( x ) . Since e q ( x ) is irreducible and S ( x ) is the square-free part of R ( x ) , we have S ( x ) = e q ( x ) and hence R ( x d ) = e q ( x d ) m for some m ∈ N [ x ] . Finally, since the roots of e q ( x d ) are Z d , the roots of e q ( x ) are { z d | q ( z ) = } . Corollary 4.8.
Let d ∈ N and f = (cid:213) mj = q a j j , where ∈ N and q j are primitive irreducible polynomials in Z [ x ] with positive leading coefficients. Let q ∗ i ( x d ) be the square-free part of R u ( u d − x d , q i ( u )) and f ∗ , lcm ( { q ∗ a j j | j } ) . Then ( f ) Z [ x ] \ Z [ x d ] = ( f ∗ ( x d )) Z [ x d ] . (14) Furthermore, the roots of f ∗ ( x ) are { z d | f ( z ) = } .Proof: By Lemma 4.7, we have ( q i ) Z [ x ] T Z [ x d ] = ( q ∗ i ( x d )) Z [ x d ] . Then ( f ) Z [ x ] T Z [ x d ] = s T i = (( q a i i ) Z [ x ] T Z [ x d ] = s T i = ( q ∗ a i i ) Z [ x d ] = ( lcm ( { q ∗ a i i | i } )) Z [ x d ] = ( f ∗ ( x d )) Z [ x d ] . From f ∗ , lcm ( { q ∗ a j j | j } ) and Lemma 4.7, the roots of f ∗ ( x ) are { z d | f ( z ) = } . Theorem 4.9.
Let f ∈ Z [ x ] have a unique positive root x + and any root w of f satisfies | w | ≤ x + . If there existsa minimal d ∈ N > such that for all root z = x + of f , | z | = x + implies z d ∈ R > . Let f ∗ ( x d ) ∈ Z [ x d ] be thepolynomial in (14). Then f ∈ F if and only if lc ( f ) = lc ( f ∗ ) and f ∗ ∈ F .Proof: “ ⇐ " Since lc ( f ) = lc ( f ∗ ) and ( f ) ∩ Z [ x d ] = ( f ∗ ( x d )) , there exists a monic polynomial h ∈ Z [ x ] suchthat f ∗ ( x d ) = f h . Since f ∗ ∈ F , there exists a monic polynomial g ∈ Z [ x ] such that f ∗ ( x ) g ( x ) ∈ F . Then f ∗ ( x d ) g ( x d ) = f hg ( x d ) ∈ F . Since hg ( x d ) is monic, we have f ∈ F .“ ⇒ " Since f ∈ F , there exists a primitive polynomial h ∈ ( f ) T F with h ( ) = ( h ) = lc ( f ) .Each such h has some roots whose absolute value is x + . Since f | h , by part 3 of Lemma 4.3 we have d | d h ,where d h = gcd { k | x k is in h } . By Lemma 4.3, h ∈ Z [ x d h ] ⊂ Z [ x d ] . Thus h ∈ ( f ) T Z [ x d ] = ( f ∗ ) Z [ x d ] . Sincelc ( f ) | lc ( f ∗ ) | lc ( h ) and lc ( f ) = lc ( h ) , we have lc ( f ) = lc ( f ∗ ) = lc ( h ) , so f ∗ ∈ F . Example 4.10.
Let f = ( x − )( x + ) . Then d = and f ∗ = ( x − )( x − ) has two positive roots and hencef F by Corollary 4.4 and Theorem 4.9. et f = x − , f = x − x + , and f = f f . Then d = , f ∗ = x − , f ∗ = x − , and f ∗ = x − .Hence f ∈ F . Corollary 4.11.
Let f ∗ ( x ) be the polynomial defined in Theorem 4.9. Then f ∗ ( x ) has only one root (may be amultiple root) whose absolute value is x d + and any root z = x d + of f ∗ satisfies | z | < x d + .Proof: By Corollary 4.8, the roots of f ∗ ( x ) are { z d | f ( z ) = } . Then the corollary comes from the fact that x + is the unique positive real root of f and f ( z ) = , | z | = x + imply z d ∈ R > .By Corollary 4.11, when f has a unique positive real root x + , we reduce the decision of f ∈ F into thedecision of f ∗ ∈ F , where f ∗ has only one root with absolute value x d + . Lemma 4.12.
If f ∈ F \ F has a unique positive real root x + , then x + > .Proof: There exists a monic polynomial g ∈ Z [ x ] such that f g ∈ F . Since f / ∈ F , g is not a monomial.Without loss of generality we assume g ( ) =
0, and then (cid:213) g ( z )= | z | = | g ( ) / lc ( g ) | = | g ( ) | ≥ g ( z )= ( | z | ) >
1. Since x + is the unique positive root of f g , by Lemma 4.2, we have x + > max g ( z )= ( | z | ) > f ∈ F in the case of f ( ) = Lemma 4.13.
Let f ∈ Z [ x ] be a primitive polynomial, f ( ) = . If d ∈ N is the smallest number such that allroot z of f satisfies z d = , then f ∈ F if and only if f ∗ ( x ) = x − , where f ∗ is defined in (14).Proof: By Theorem 4.9, if f ∗ ( x ) = x − f ∈ F . Suppose f ∈ F . By Lemma 4.3, any root of f is simpleand hence f is square-free. Let d = lcm { m ∈ N | z m = } . Since f is primitive, d ∈ N is the smallest numbersuch that f ( x ) | x d − Z [ x ] . Therefore, so f ∗ ( x ) = x − Example 4.14.
Let f = ( x − )( x + )( x + ) . Then d = and f ∗ = x − . So, f ∈ F . Let f = ( x − )( x + ) ( x + ) . Then d = and f ∗ = ( x − ) . So, f / ∈ F . Lemma 4.15.
If f ( ) = and any other root z of f satisfies | z | < , then f ∈ F if and only if f ( x ) / ( x − ) ∈ Z [ x d ] for some d ∈ N > and f ( x )( x d − ) / ( x − ) ∈ F .Proof: The necessity is obvious. For the other direction, there exists a monic polynomial g ∈ Z [ x ] such that f g ∈ F . We claim that each root z of g has absolute value 1. Since g is monic, (cid:213) g ( z )= | z | ≥
1. Since f g ∈ F and f ( ) =
0, max g ( z )= | z | ≤
1, and the claim is proved.By Lemma 4.2, f g ∈ Z [ x d ] , where d = d f g . Since f ( ) = f have absolute value <
1, we have ( x d − ) | f g and (( x d − ) / ( x − )) | g . By part 3 of Lemma 4.3, the roots of f g with absolutevalue 1 are exactly the roots of x d −
1. Since the absolute values of all roots of g is 1 and g has no multipleroots by Lemma 4.3, g = ( x d − ) / ( x − ) . Since f g ∈ Z [ x d ] and ( x d − ) | f g , set f g = ( x d − ) h ( x d ) for h ∈ Z [ x ] . From g = ( x d − ) / ( x − ) , we have f / ( x − ) = h ( x d ) ∈ Z [ x d ] .Now, only when f / ∈ F , f has a unique positive real root x + >
1, and any other root of f has absolutevalue < x + , we do not know how to decide f ∈ F . By computing many examples, we propose the followingconjecture. Conjecture 4.16.
If f ∈ Z [ x ] \ F has a simple and unique positive real root x + , x + > , and x + > | z | for anyother root z of f , then f ∈ F . f ∈ F Based on the results proved in the preceding section, we give the following algorithm to decide whether f ∈ F .Note that the last step of the algorithm depends on whether Conjecture 4.16 is true.12 lgorithm 1 — Membership F ( f ) Input: f ∈ Z [ x ] such that lc ( f ) > f ( ) =
0, and f is primitive. Output:
Whether f ∈ F .1. If lt ( f ) = f + , then f ∈ F ⊂ F .2. If f has no positive real roots, then f ∈ F .3. If f has at least two positive real roots (with multiplicities counted), then f / ∈ F .4. Let x + be the simple and unique positive real root of f .4.1. If x + <
1, or equivalently f ( ) > f / ∈ F .4.2. If x + = z of f satisfies z d = d ∈ N , then f ∈ F if and only if f ∗ = x − f ∗ is defined in (14).4.3. If x + = z of f satisfies | z | <
1, then f ∈ F if and only if f ( x ) / ( x − ) ∈ Z [ x d ] forsome d ∈ N > and f ( x )( x d − ) / ( x − ) ∈ F .4.4. If f has a root z such that | z | > x + , then f / ∈ F .4.5. If f has a root z such that z = x + , | z | = x + , and ( zx + ) d = d ∈ N > , then f / ∈ F .4.6. Let d be the minimal integer such that f ( z ) = z = x + , and | z | = x + imply ( zx + ) d =
1. Then f ∈ F if and only if lc ( f ) = lc ( f ∗ ) and f ∗ ∈ F , where f ∗ is defined in (14). If lc ( f ) = lc ( f ∗ ) then return Membership F ( f ∗ ) , otherwise return false.4.7. If f does not satisfy all the above conditions, then it satisfies the condition of Conjecture 4.16 and f ∈ F if the conjecture is valid.In what below, we will give the details for Algorithm 1 and prove its correctness. We will use algorithmsfor real root isolation and complex root isolation for univariate polynomials. Please refer to the latest work onthese topics and references in these papers [2, 19].Step 1 is trivial to check. Step 2 can be done with any real root isolation algorithm. Step 3 can be done byfirst factoring f as the product of irreducible polynomials and then isolating the real roots of each factor of f .Step 4.1 is trivial to check. For Step 4.2, there exists a d ∈ N such that ( z ) d = f ( x ) is a cyclotomic polynomial, which can be checked with the Graeffe method in [3]and the d can also be founded. The polynomial f ∗ in Step 4.2 can be computed with Corollary 4.8.In Step 4.3, the d can be found from the fact f ( x ) / ( x − ) ∈ Z [ x d ] . If f ( x )( x d − ) / ( x − ) ∈ F for some d satisfying f ( x ) / ( x − ) ∈ Z [ x d ] , then return true; otherwise return false.In Steps 4.4, 4.5, and 4.6, we need to check whether f has a root z = x + such that | z | > x + , | z | = x + , and z m ∈ R > for some m ∈ N . To do that, we first give a lemma. Lemma 4.17.
Let p ( x ) = a (cid:213) ni = ( x − x i ) ∈ Z [ x ] , q ( x ) = b (cid:213) mj = ( x − y j ) ∈ Z [ x ] , and x i y j = for all i , j. Thenthe roots of R u ( p ( u ) , q ( ux )) are { y j / x i | i = , · · · , n , j = , · · · , m } and the roots of R u ( u n p ( x / u ) , q ( u )) are { x i y j | i = , · · · , n , j = , · · · , m } .Proof: The lemma comes from R u ( p ( u ) , q ( ux )) = a m b n (cid:213) i , j ( x − x j / y i ) and R u ( u n p ( x / u ) , q ( u )) = a m b n (cid:213) i , j ( x − x i x j ) , where a = p ( ) .In the rest of this section, we assume f = f t (cid:213) i = f e i i r i ( x ) = R u ( u n f i ( x / u ) , f i ( u )) (15)13here f i are primitive and irreducible polynomials with positive leading coefficients. Also assume that f ( x ) has a unique positive root x + which is the root of f ( x ) .By Lemma 4.17, the real roots of all r i ( x ) include x + and zz , where z is a complex root of r i ( x ) . Then thecondition in Step 4.4 of the algorithm can be checked with the following result based on real root isolation. Corollary 4.18. f has a root z such that | z | > x + if and only if some r i ( x ) has a positive root larger than x + . It is easy to check whether − x + is a root of f i : since f i is irreducible, − x + is a root of f i if and only if f i ( − x ) = ± f i ( x ) . If z is complex root of f i such that | z | = x + , then x + , x + = z . z , x + = z . z are all roots of r i .Then, we have the following result. Corollary 4.19.
Let m i be the multiplicity of x + as a root of r i and n i the multiplicity of − x + as a root of f i (themultiplicity is set to be zero if x + or − x + is not a root). Then { z | f ( z ) = , | z | = x + , z / ∈ R } = m − n − and { z | f i ( z ) = , | z | = x + , z / ∈ R } = m i − n i for i > . As usual, a representation of a complex root z is a pair ( p , B ) where p is an irreducible polynomial and B a box such that p ( z ) = z is the only root of p in B . A box is represented by its lower-left and upper-rightvertexes: ([ x l , y l ] , [ x t , x t ]) . By the following lemma, we can find representations for all roots z of f satisfying | z | = x + . Lemma 4.20.
Suppose f i has s roots z , . . . , z s satisfying | z j | = x + . Then, we can find representations for z j .Proof: Since f i is irreducible, f i is the minimal polynomial for z i . Suppose I = ( a , b ) is an isolation intervalfor x + . By algorithms of complex root isolation and real root isolation, we can simultaneously refine I andthe isolation boxes of the roots of f i such that the number of isolation boxes meet the region a < | x | < b willeventually becomes s . These s boxes are the isolation boxes for z , . . . , z s , since f i has exactly s roots satisfying | z | = x + . Lemma 4.21.
Let z be a root of f k satisfying | z | = x + . Then, we can find a representation for z / x + .Proof: Let H ( x ) = R u ( f ( u ) , f k ( ux )) ∈ Z [ x ] and h i ( x ) , i = , . . . , s the irreducible factors of H . From Lemma4.17, H ( z / x + ) = h c ( z / x + ) = c and we will show how to find h c . Isolate the roots of h i , i = , . . . , s and refine the isolation box B = ([ x l , y l ] , [ x t , x t ]) of z and the isolation interval of x + = ( l , r ) simultaneously such that ([ x l / r , y l / r ] , [ x t / l , x t / l ]) intersects only one of the isolation boxes of h i , i = , . . . , s .This box B should be the isolation box for z / x + . If B contains a root of f c , then f c is the minimal polynomialfor z / x + .With the following lemma, we can check whether z m ∈ R > for some m . Lemma 4.22.
Let z be a root of f k satisfying | z | = x + and q the minimal polynomial for z / x + . Then we candecide whether there exists an m ∈ N such that ( z / x + ) m = , and if such an m exists, we can compute theminimal m.Proof: There exists an m ∈ N such that ( z / x + ) m = q ( x ) is a cyclotomic polynomial, which wecan be tested by the Graeffe method in [3]. The method also gives the m such that ( z / x + ) m =
1. The minimal m can be found easily.Now, we consider Step 4.5. With Corollary 4.19 and Lemma 4.20, we can find all the roots z of f satisfying | z | = x + . For each such z , we can check whether there exists a d z ∈ N such that ( z / x + ) d z = ?? .Hence the conditions of Step 4.5 can be checked.Now, we consider Step 4.6. The d in Step 4.6 can be computed as d = lcm { d z | f ( z ) = , | z | = x + , ( z / x + ) d z = } . With d given, f ∗ in Step 4.6 can be computed with Corollary 4.8. From Corollary 4.8, the roots of f ∗ are { z d | f ( z ) = } . As a consequence, when running Membership F ( f ∗ ) , only Steps 1, 3, 4.7 will be executed,and no further calls to Membership F ( f ∗ ) are needed.14 .3 Compute the finite s -Gröbner basis Let f ∈ F , we will show how to compute the finite s -Gröbner basis for I f = sat ( P f ) in (6). Lemma 4.23.
Let f ∈ F , h = f g ∈ F for a monic g ∈ Z [ x ] , and D = deg ( h ) . Then I D = sat ( P f ) \ F [ y , y x , · · · , y x D ] = asat ( P f , P x f , . . . , P x D − deg ( f ) f ) (16) and a Gröbner basis of I D is a s -Gröbner basis of I f .Proof: By the remark before Theorem 2.6, P f is regular and coherent. Then P ∈ I D if and only if prem ( P , P f ) = P ∈ asat ( P f , P x f , . . . , P x D − deg ( f ) f ) [7], and (16) is proved. By Corollary 3.4, a Gröbnerbasis of I D is a s -Gröbner basis of I f .The Gröbner basis of I D , denoted as G ( f , D ) , can be computed with the following well-known factasat ( P f , P x f , . . . , P x D − deg ( f ) f ) = ( z · J (cid:229) D − deg ( f ) i = x i − , P f , P x f , . . . , P x D − deg ( f ) f ) ∩ F [ y , y x , · · · , y x D ] , where J = init ( P f ) and z is a new indeterminate. Therefore, in order to compute the s -Gröbner basis of I f , itsuffices to compute D . We thus have the following algorithm. Algorithm 2 — FiniteGB ( f ) Input: f ∈ F such that lc ( f ) > Output:
Return s -Gröbner basis of I f = sat ( P f ) .1. If lt ( f ) = f + , then return { P f } .2. If f has no positive real roots, then return G ( f , N f + deg ( f ) + ) , where N f is defined in Lemma 3.14.3. Let x + be the unique simple positive real root of f .3.1. If x + = z of f satisfies z d = d ∈ N , then return G ( f , d ) .3.2. If x + = z of f satisfies | z | <
1, then return G ( f , deg ( f ) + d − ) , where d is foundin Step 4.3 of Algorithm 1.3.3. Let d be the minimal integer such that f ( z ) = z = x + , and | z | = x + imply ( zx + ) d =
1. Let the f ∗ bedefined (14) and f ∗ ( x d ) = f ( x ) s ( x ) . Return G ( f , d deg ( f ∗ )) .In the rest of this section, we will prove the correctness of the algorithm. Step 1 follows Lemma 3.2.For Step 2, by Lemma 3.14, ( x + ) N f f ∈ Z > [ x ] . Following the proof of Lemma 3.6, for a sufficientlylarge M ∈ N , ( x − M )( x + ) N f f ∈ F . Then, D = deg (( x − M )( x + ) N f f ) = N f + deg ( f ) + f ∗ ( x d ) = f ( x ) g ( x ) = x d − g . Then D = d . For Step 3.2, following Step 4.3 of Algorithm 1, f ( x )( x d − ) / ( x − ) ∈ F . Then D = deg ( f ) + d − f ∗ ( x ) ∈ F , f ∗ ( x ) has at least two positive roots, or f ∗ satisfies the conditions of Conjecture 4.16. Since we already assumed f ∗ ∈ F , only f ∗ ( x ) ∈ F is possible. From f ∗ ( x d ) = f ( x ) s ( x ) , we have D = d deg ( f ) . We now proved thecorrectness of Algorithm 2. Given an f ∈ Z [ x ] , the existence of a monic polynomial g ∈ Z [ x ] with deg ( g ) ≤ m , such that f g ∈ F can bereduced to an integer programming problem. Based on this idea, a lower bound for deg ( g ) is given in certaincases. 15 emma 5.1. Given a polynomial f ( x ) = a n x n + · · · + a ∈ Z [ x ] with a n > , there exists a monic polynomialg ∈ Z [ x ] with deg ( g ) ≤ m, such that f g ∈ F if and only if a ( b m − , · · · , b ) ∈ Z m satisfies a n − a n ... ... . . . a a · · · a n . . . . . . . . . . . . a a · · · a n . . . . . . ... a a a ( m + n ) × ( m + ) b m − b m − ... b ≤ . (17) Moreover such g has degree < m if and only if b = for some feasible solution of the above inequalities.Proof: Let g ( x ) = x m + b m − x m − + · · · + b . The leading coefficient of f g is a n >
0, and the coefficient of x k is the m + n − k -row of the left side of (17) for k = m + n − , . . . ,
0. If deg ( g ) < m , the coefficients of g ( x ) = x m − deg ( g ) g ( x ) is a feasible solution with b =
0. If b = ( , b m − , · · · , b ) is a feasible solution of (17)for m = m − g . Lemma 5.2.
Given a polynomial f ( x ) = a n x n + · · · + a ∈ Z [ x ] with a n > , let ( / f )( x ) , l + · · · + l m x m + · · · ∈ Z [ a − ][[ x ]] . There exists a monic polynomial g ∈ Z [ x ] with deg ( g ) ≤ m and f g ∈ F if and only if thereexists a ( c m + n − , · · · , c ) ∈ N m + n such that l l l · · · l m + n − l m + n − l l . . . . . . l m + n − . . . . . . . . . ... l · · · l m c m + n − c m + n − ... c = ... − . (18) Proof:
Extending the proof of Lemma 5.1, let b m + n − , ( b m + n − , · · · , b ) T . For the following special Jordanform J j , . . . . . .. . . j × j , we have f ( J j ) = a · · · a n a . . . . . .. . . . . . a n a ... a j × j . By Lemma 5.1, f g ∈ F if and only if f ( J m + n ) b ∈ Z m + n ≤ for some ( b m − , · · · , b ) ∈ Z m with ( b m + n − , · · · , b m ) =( , · · · , , ) . Let c = ( c m + n − , · · · , c ) T , − f ( J m + n ) b ∈ N m + n . Then we have f ( J m + n ) − c = ( / f )( J m + n ) c = b , that is l l · · · l m + n − l . . . .... . . l l c m + n − c m + n − ... c = ... − − b m − ... − b . Since we need only to know the existence of c i , only the first n rows are need, and the lemma is proved.Note that a i + l i ∈ Z for any i ∈ N . We can reduce the coefficient matrix in the above lemma into an integermatrix. Corollary 5.3.
Let f , g ∈ R [ x ] , lc ( f ) > , g monic, and ( / f )( x ) , (cid:229) ¥ m = l m x m ∈ R [[ x ]] . If lt ( f g ) = ( f g ) + ,then deg ( g ) ≥ min { j ∈ N | l j < } .Proof: From the proof of Lemma 5.2, there exists a monic g ∈ R [ x ] such that lt ( f g ) = ( f g ) + if and only if(18) has a solution ( c m + n − , · · · , c ) ∈ R m + n > . If l , . . . , l m ≥
0, the last coordinate of (18) is (cid:229) mj = l j c m − j ≥
0, hence (cid:229) mj = l j c m − j = − R m + n > . As a consequence, if lt ( f g ) = ( f g ) + , thendeg ( g ) ≥ min { j ∈ N | l j < } and the corollary is proved. Corollary 5.4.
Let f ( x ) = ax + bx + c ∈ R [ x ] , a > , b − ac < , and z a root of f . If f g ∈ F and g ismonic, then deg ( g ) ≥ ⌊ p / | Arg ( z ) |⌋ = ⌊ p / arctan ( √ ac − b / b ) ⌋ .Proof: Let f ( x ) = a ( x − z )( x − ¯ z ) , and z = re q i where r ∈ R > and q = Arg ( z ) = k p . Without loss of generality,we can assume 0 < q < p . Then1 f ( x ) = a ( x − z )( x − ¯ z ) = ¥ (cid:229) j = z j + − ¯ z j + a ( z ¯ z ) j + ( z − ¯ z ) x j = ¥ (cid:229) j = sin (( j + ) q ) ar j + sin q x j , that is, l j = sin (( j + ) q ) ar j + sin q . Since l = ar >
0, min { j ∈ N | l j < } = min { j ∈ N | ( j + ) q > p } = ⌊ p / q − ⌋ + = ⌊ p / q ⌋ . By Corollary 5.3, deg ( g ) ≥ ⌊ p / q ⌋ = ⌊ p / arctan ( √ ac − b / b ) ⌋ .We can now give a lower bound for the degree of g such that f g ∈ F in certain case. Theorem 5.5.
If a polynomial f ( x ) ∈ Z [ x ] is of degree n and has at least one root not in R , then min { deg ( g ) | g ∈ Z [ x ] is monic and f g ∈ F } ≥ max {⌊ p / | Arg ( z ) |⌋ − n + | f ( z ) = , z / ∈ R } .Proof: Since f ( x ) ∈ Z [ x ] has at least one root not in R , f = f f where f is a quadratic polynomial in R [ x ] which has two complex roots. Suppose there exists a monic g ∈ R [ x ] such that lt ( f g ) = ( f g ) + or lt ( f f g ) =( f f g ) + . By Corollary 5.4, deg ( g ) ≥ ⌊ p / | Arg ( z ) |⌋ − deg ( f ) = ⌊ p / | Arg ( z ) |⌋ − n +
2. Then, min { deg ( g ) | g ∈ Z [ x ] is monic and f g ∈ F } ≥ min { deg ( g ) | g ∈ R [ x ] is monic and lt ( f g ) = ( f g ) + } ≥ max {⌊ p / | Arg ( z ) |⌋ − n + | f ( z ) = , z / ∈ R } .The following result shows that the lower bound given in the preceding theorem is also the upper boundfor quadratic polynomials. Proposition 5.6.
Let f ( x ) = a x + a x + a = a ( x − z )( x − ¯ z ) be a quadratic polynomial in Z [ x ] with a rootcomplex z = a + b i = re q i , where a , b , r > , < q < p , ¯ z = a − b i . Then min { deg ( g ) | g ∈ Z [ x ] and monic , f g ∈ F } = ⌊ p / q ⌋ . roof: If p / < q < p , then a = − a > f ∈ N > [ x ] . By the proof of Lemma 3.6, there exists an N such that ( x − N ) f ∈ F and hence c deg ( f ) = = ⌊ p / q ⌋ . If q = p /
2, then f = a x + a . It is easy to check c deg ( f ) = = ⌊ p / q ⌋ .From now on, we assume 0 < q < p /
2, so a > a <
0. Considering f ( x ) = ( x − a − bi )( x − a + bi ) = x − ax + a + b ∈ Z [ a − ][ x ] , we will solve the integer programming mentioned in Lemma 5.1: − a a + b − a . . . . . . . . . a + b − a a + b − aa + b ( m + ) × ( m + ) b m − b m − ... b ≤ . (19)Let D = − a and r D j + = − a − ( a + b ) / D j for j >
1. Then D j = − ( a + bi ) j + − ( a − bi ) j + ( a + bi ) j − ( a − bi ) j = − r sin ( j + ) q sin j q . Let m = ⌈ p / q ⌉ −
1. Then we have D j < j = , · · · , m − D m ≥ m = m −
1. We add ( m + ) -th rowmultiplied by 1 / ( − D ) > m -th row. Then the − a at the m -th row becomes D = − a − ( a + b ) / D ,and the 1 at the m -th row becomes 0. Then add m -th row multiplied by 1 / ( − D ) > ( m − ) -th row.Repeat the above process until D m ≥
0, and we obtain a lower triangular matrix: D m a + b D m − . . . . . . . . . a + b D a + b D a + b m + × m . (20)1. If D m >
0, the first coordinate of the left side of D m a + b D m − . . . . . . a + b D a + b b m − b m − ... b ≤ D m >
0. So the feasible region of (21) is empty and hence the feasible region of (19) is also empty.Thus f g / ∈ F for any monic polynomial g of degree < m by Lemma 5.1.Let m = m . We have − a a + b D m a + b D m − . . . . . . a + b D a + b b m − b m − ... b ≤ . (22)18imilarly, we can obtain a quasi-upper trangular matrix from (19) by row transformations: D . . . . . . D m − D m a + b − a b m − b m − ... b ≤ . (23)Combining (19), (22) and (23), we have b m − ≤ − a + b D m ⇒ b m − < ⇒ − a + b m − ≤
0; (24) − a + b D j + b j + ≤ b j ≤ − ( a + b ) b j + + ab j + , j = m − , m − , · · · ,
0; (25) b j ≤ − ( a + b ) b j + + ab j + ⇒ b j ≤ − D m − j b j + ⇒ b j < , j = m − , m − , · · · ,
1; (26) b ≤ ⇒ D m b + b ≤ . (27)In (25), we need to show that there exists a rational number b j satisfying − a + b D j + b j + < b j < − ( a + b ) b j + + ab j + . (28)We need to show − ( a + b ) b j + + ab j + + a + b D j + b j + = − ( a + b ) b j + − D j + b j + > , which is true from the first ‘ < ’ in (28) when j = j + b m − , · · · , b satisfying (24) and (28), and then ( , b m − , · · · , b ) is a feasible solution of (19). Taking the common denominator N ∈ N ≥ of { b j | j = , · · · , m − } ,we have − a + Nb m − < − a + b m − ≤ a + b − aNb m − + Nb m − < N ( a + b − ab m − + b m − ) ≤ ( a + b ) Nb j − aNb j − + Nb j − ≤ , j = m − , · · · , ( a + b ) Nb − aNb ≤ ( a + b ) Nb ≤ , and then f ( x ) g ( x ) = a ( x − ax + a + b )( x m + m − (cid:229) j = Nb j x j ) ∈ F . (29)Then D m > c deg ( f ) = m = ⌈ p / q ⌉ − = ⌊ p / q ⌋ .19. If D m = p / q = m + > z = re p i / ( m + ) . Then e p i / ( m + ) is a root of ( x − ) − R u ( f ( x ) , f ( ux )) = a a x + ( a a − a ) x + a a . Since e p i / ( m + ) is integral over Z , we have a a | ( a a − a ) or a a | a . For 0 < p / ( m + ) < p , a a x + ( a a − a ) x + a a has no real roots, and then we have ( a a − a ) − ( a a ) <
0, that is a < a a . Then we have m = a = a a , m = a = a a or m = a = a a .(a) If m = D = f ( x ) = a x + a x + a , where a = −√ a a . When solving (19) for m = b ≤ a a , b ≤ − a b a , − a + a b a a ≤ b ≤ − a b + a a b a . In order for an integer b to satisfy these inequations, we need to assume a b − a a b a + a + a b a a ≥ , that is b ≤ a − a a a . Here b < { deg ( g ) | f g ∈ F } ≥
3, so c deg ( f ) = = p / q = ⌊ p / q ⌋ .(b) If m = D = f ( x ) = a x + a x + a , where a = −√ a a . When we solve (19) for m =
4, we have b ≤ − a a , b ≤ a a + a a b − a a , − a b + a a b − a a ≤ b ≤ − a − a b a , − a b − a b a ≤ b ≤ − a b − a b a . When we want − a b − a b a − − a b − a b a ≥ , − a − a b a − − a b + a a b − a a ≥ , we only need b ≤ min { − a + a a a , − a − a a a } , b ≤ a a + a a b − a a . Here b ≤ − a / a < { deg ( g ) | f g ∈ F } ≥
4, so c deg ( f ) = = p / q = ⌊ p / q ⌋ .(c) If m = D = f ( x ) = a x + a x + a , where a = −√ a a . Rewriting a f ( x ) = a x + a a x + a , When we solve (19) for a f ( x ) for m =
6, we get b < , b ≤ − a + a a b a , a b a ≤ b ≤ a a b − a b a a b a ≤ b ≤ a a b − a b a , a b a ≤ b ≤ a a b − a b a , a b a ≤ b ≤ a a b − a b a . Because b < a b a < a a b − a b a , a b a < b implies a b a < a a b − a b a , a b a < b im-plies a b a < a a b − a b a , and a b a < b implies a b a < a a b − a b a , there exists a feasible solu-tion { b , b , b , b , b , b } ∈ Q < , which is an inner point of the semi-algebraic set. Using thesame notations in (29), let N ∈ N > be the common denominator of { b , . . . , b } , and we have f ( x )( x + N (cid:229) j = b j x j ) ∈ F .Here b < { deg ( g ) | f g ∈ F } ≥
6, so c deg ( f ) = = p / q = ⌊ p / q ⌋ .20e complete the proof.The following example is used to illustrate the proof. Example 5.7.
Let f = x − x + , D = − , D = > , m = , c deg ( f ) = . Here f / ∈ N [ x ] implies c deg ( f ) > ,and ( x − x + )( x − x − ) ∈ F implies c deg ( f ) ≤ . Example 5.8.
Let f = x − x + . By the effective Polya Theorem 3.14, we have d = min { deg ( g ) | g ∈ Z [ x ] and monic , f g ∈ F } ≤ . However, we have min { deg ( g ) | g ∈ Z [ x ] and monic , f g ∈ F } = by propo-sition 5.6, where g = x − x − x − and f g = x − x − . In this paper, we study when a s -ideal has a finite s -Göbner basis. We focused on a special class of s -ideals: normal binomial s -ideals which can be be described by the Gröbner basis of a Z [ x ] -module. We give acriterion for a univariate normal binomial s -ideal to have a finite s -Gröbner basis. When the characteristic setof the s -ideal consists of one s -polynomial, we can give constructive criteria for the s -ideal to have a finite s -Gröbner basis and an algorithm to compute the finite s -Gröbner basis under these criteria. One case is stillnot solved and we summary it as a conjecture. Also, it is desirable to extend the criteria given in this paperto multivariate binomial s -ideals. Example 2.9 shows that extending Theorem 3.1 to the multivariate case isquite nontrivial. For s -Gröbner basis of general s -ideals, the work on monomial s -ideals may be helpful [20]. References [1] G.E. Collins and A.G. Akritas. Polynomial Real Root Isolation Using Descarte’s Rule of Signs,
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