It is well-known that the resultant, as a basic concept in algebraic geometry and a powerful tool in elimination theory, gives conditions for an over-determined system of polynomial equations to have common solutions . Since most polynomials are sparse in that they only contain certain fixed monomials, Gelfand, Kapranov, Sturmfels, and Zelevinsky introduced the concept of sparse resultant [5, 6]. Later various effective algorithms are proposed to compute sparse resultant. In particular, Canny and Emiris showed that the sparse resultant is a factor of the determinant of a Macaulay style matrix and gave an efficient algorithm to compute the sparse resultant based on this matrix representation . Andrea further showed the sparse resultant is the quotient of two determinants where the denominator is a minor of the numerator .
With the resultant and sparse resultant theories becoming more mature, extending the algebraic results to differential and difference cases is a natural way. However, such results in differential and difference cases are not complete parallel with algebraic case, even not hold. Since our paper concentrates on the difference case, though differential resultant and sparse differential resultant are studied successively by many researchers, we will not state them in detail and refer to [13, 12, 11] and references therein.
For the difference case, Li, Yuan and Gao introduced the concept of sparse difference resultant for a Laurent transformally essential system consisting of Laurent difference polynomials in difference variables and its basic properties are proved . Based on the degree and order bound, they proposed a single exponential algorithm in terms of the number of variables, the Jacobi number, and the size of the Laurent transformally essential system, which is essentially to search for sparse difference resultant with order and degree bound.
In this paper, we further explore efficient algorithms to find the sparse difference resultant of the given difference polynomial system. We show that the sparse difference resultant of a Laurent transformally essential system consisting of Laurent difference polynomials in difference variables is the same to the one of a simple system consisting of polynomials in difference variables, where is the rank of the symbolic support matrix of the super essential system. Moreover, new order bound of sparse difference resultant is given. Then we propose an efficient algorithm to compute sparse difference resultant, which is based on the results that sparse difference resultant is shown to be the algebraic sparse resultant for a certain generic polynomial system. It starts with the given sparse difference polynomial system and directly obtain a strong essential polynomial system of the original system, then one can regard it as sparse algebraic polynomial system and use the algorithm in  to construct the matrix representation whose determinant is the required sparse difference resultant. Furthermore, the whole computations of the algorithm are compiled to the function SDResultant implemented with Mathematica.
The rest of the paper is arranged as follows. In Section 2, we review some preliminary results which contains definitions and theorems of sparse resultant and sparse difference resultant. Section 3 concentrates on the main results of the paper involving the theoretical preparation of the algorithm, algorithm implementation and an illustrated example. The last section concludes the results.
2.1 Sparse resultant
Let be finite subsets of . Assume and for each . For algebraic indeterminates and , denote . Let
be generic sparse Laurent polynomials. We call the support of and is called the symbolic support vector of . The smallest convex subset of containing is called the Newton polytope of . For any subset , the matrix
whose row vectors areis called the symbolic support matrix of . Denote , and by the rank of matrix .
Follow the notations introduced above.
A collection of is said to be weak essential if .
A collection of is said to be essential if and for each proper subset J of I, .
A polynomial system is weak essential if and only if is of codimension one. In this case, there exists an irreducible polynomial such that and is called the sparse resultant of . Furthermore, the system is essential if and only if and appears effectively in for each .
Suppose an arbitrary total ordering of is given, say . Now we define a total ordering among subsets of . For any two subsets and where and , is said to be of higher ranking than , denoted by , if 1) there exists an such that , or 2) and . Note that if is a proper subset of , then . Thus for the system given in (1), if , then has an essential subset with minimal ranking.
Suppose is an essential system. Then there
exists an with , such that by setting to , the specialized system
is still essential.
is the number of variables in .
, where .
An essential system is said to be variable-essential if there are only variables appearing effectively in . Clearly, if is essential, then it is variable-essential.
We call a variable-essential system strong essential if also satisfies condition (2) in Lemma 2.3.
2.2 Sparse difference resultant
This section will review the results associated with sparse difference resultant, for details please refer to reference .
An ordinary difference field is a field with a third unitary operation satisfying that for any , , , and if and only if . We call the transforming operator of . If , is called the transform of and is denoted by . And for , is called the -th transform of and denoted by , with the usual assumption . By we mean the set . If is defined for each , is reflexive. Every difference field has an inversive closure . In this paper, all difference fields are assumed to be inversive with characteristic zero. A typical example of reflexive difference field is with .
A subset of a difference extension field of is said to be transformally dependent over if the set is algebraically dependent over , otherwise, is called transformally independent over , or a family of difference indeterminates over . In the case consists of one element , we say that is transformally algebraic or transformally transcendental over , respectively. The maximal subset of which are transformally independent over is said to be a transformal transcendence basis of over . We use to denote the difference transcendence degree of over , which is the cardinal number of .
For every Laurent difference polynomial , there exists a unique laurent difference monomial such that is the norm form of , denoted by N, which satisfies 1) and 2) for any Laurent difference monomial with , is divisible by as polynomials. The order and degree of is defined to be the order and degree of , denoted by and .
Let be an ordinary difference field with a transforming operator and the ring of difference polynomials in the difference indeterminates . Let be a difference polynomial in . The order of w.r.t. is defined to be the greatest number such that appears effectively in , denoted by . If does not appear in , then we set . The order of is defined to be , that is, . A Laurent difference monomial of order is in the form where are integers which can be negative. A Laurent difference polynomial over is a finite linear combination of Laurent difference monomials with coefficients in .
Suppose are finite sets of Laurent difference monomials in Consider generic Laurent difference polynomials defined over :
where all the are transformally independent over . Denote
The number is called the size of and is called the support of . To avoid the triviality, are always assumed in this paper.
A set of Laurent difference polynomials of the form (2) is called Laurent transformally essential if there exist with such that In this case, we also say that form a Laurent transformally essential system.
Let be the set of all difference monomials in and the difference ideal generated by in . Let
Now suppose is a Laurent transformally essential system. Since defined in (5) is a reflexive prime difference ideal of codimension one, there exists a unique irreducible difference polynomial such that can serve as the first polynomial in each characteristic set of w.r.t. any ranking endowed on . Thus the definition of sparse difference resultant is given as follows:
The above is defined to be the sparse difference resultant of the Laurent transformally essential system , denoted by or , where is the support of for . When all the are equal to the same , we simply denote it by .
The first step to find sparse difference resultant of a Laurent difference polynomial system is to determine whether the system is Laurent transformally essential or not. In , the authors build a one-to-one correspondence between a difference polynomial system and so-called symbolic support matrix and then use the matrix to check whether a Laurent difference system is transformally essential.
Specifically, let be Laurent difference monomials. Introduce a new algebraic indeterminate and let
be univariate polynomials in . If , then set . The vector is called the symbolic support vector of . The matrix is called the symbolic support matrix of .
Consider the set of generic Laurent difference polynomials defined in (2):
Let be the symbolic support vector of . Then the vector is called the symbolic support vector of and the matrix whose rows are is called the symbolic support matrix of . Therefore, we have
() A sufficient and necessary condition for form a Laurent transformally essential system is the rank of is equal to .
Furthermore, we can use the symbolic support matrix to discriminate certain such that their coefficients will not occur in the sparse difference resultant, which leads to the following definition:
Let . Then we call or super-essential if the following conditions hold: (1) and (2) for each proper subset of .
The existence of super-essential system of a difference polynomial system is given by the following theorem .
If is a Laurent transformally essential system, then for any , and there exists a unique which is super-essential.
Now, we introduce some notations which is needed to bound the order of . Let be an matrix where is an integer or . A diagonal sum of is any sum with a permutation of . If is an matrix with , then a diagonal sum of is a diagonal sum of any submatrix of . The Jacobi number of a matrix is the maximal diagonal sum of , denoted by .
Let and . We call the matrix the order matrix of . By , we mean the submatrix of obtained by deleting the -th row from . We use to denote the set and by , we mean the set . We call the Jacobi number of the system , also denoted by .
Let be a Laurent transformally essential system and the sparse difference resultant of . Then
3 Main results
In this section, we first present some theoretical results, and then give an efficient algorithm to compute sparse difference resultant. We analyze the algorithm complexity and present an example to illustrate the algorithm.
3.1 Theoretical preparations
Without loss of generality, we assume which is super essential, where . The symbolic support matrix of is
and . Then we choose a submatrix from whose column rank is . Without loss of generality, we assume that the first columns in is of rank . Now, we set , to in to obtain a new difference polynomial system whose symbolic support matrix is
Obviously is Laurent transformally essential since . Let be the symbolic support vector of for in , and be the sparse difference resultant of the system . We will show that .
forms a super essential system.
Proof: By Theorem 2.6, is a transformally essential system since the rank of is . Assuming that is not a super essential system, by Theorem 2.8, there exists such that is super essential, which means that the symbolic support vectors of are linear dependent. Let be the symbolic support vector of and thus , where for any .
On the other hand, since is super essential, we have , where for any , which implies . Since , we find that, for , satisfy two linear independent relations over , which means . It contradicts to . Thus forms a super essential system.
Suppose that is an algebraically essential system in the form
If is algebraically essential, then forms an algebraically essential system.
Proof: Since is an algebraically essential system, we have that the co-row rank of its symbolic support matrix is . Since is algebraically essential, we have where for any and is the symbolic support vector of , or equivalently, where is the symbolic support vector of .
Assume that does not form an algebraically essential system, then its symbolic support matrix has full rank, thus we have that where is the symbolic support vector of , or equivalently, where is the symbolic support vector of . Now we consider the rank of . By Lemma 3.1, is super essential, each sub-matrix of is of full rank. Since and , through a row transformation for over , we may obtain a row vector of form and are not all zeros. Hence, the rank of is . This contradicts to the fact that is a Laurent transformally essential system. Hence, forms an algebraically essential system.
With above notations, up to a sign.
Proof: By the definition of , we have that is the sparse resultant of . By Lemma 3.2, is an algebraically essential system. Let be the sparse resultant of , then since is obtained by setting to in for . Hence and . Then, up to a sign since they are irreducible.
Now, we show that up to a sign. By the definition of , has the lowest rank in for each . We claim that has the lowest rank in for each . If it is not the case, then has lower rank than in for some . Since , then . It contradicts to the fact that has the lowest rank in for each . Then by the uniqueness of sparse difference resultant, up to a sign.
Now we give some new order bounds for the sparse difference resultant which is obviously true according to the above results. Let be the order matrix of the system .
The order bound of for each set can be bounded by , where is any full rank sub-matrix of .
By above proposition, one may take the order bound by the minimal Jacobi number of the corresponding full rank sub-matrices of . Furthermore, we have
Let be an full rank sub-matrix of . Let be the greatest common divisor of the -th column of for . Then the order bound of for each set can be bounded by .
Proof: Let be a super essential system and its symbolic support matrix. We denote by be the -th column of and . Then and are linearly dependent. Let be the matrix by adding a last column to and its corresponding difference system. Then by Theorem 3.3, and have the same sparse difference resultant. Let be the sub-matrix of by deleting the first column and its corresponding difference system. By Theorem 3.3 again, and have the same sparse difference resultant. Inductively, one may take an matrix with as its -th column and its corresponding difference system , such that and have the same sparse difference resultant. Thus the order bound of for each set equals the -th Jacobi number of the order matrix , which is where .
In what follows, we state a proposition which will accelerate the algorithm to search for a simple algebraic polynomial system induced by .
Let be a polynomial system obtained from a transformally essential difference system, the algebraic symbolic support matrix of , Jacobi numbers are given as above, then its algebraic essential system is contained in , where .
Proof: Let and the algebraic symbolic support matrix of . We only need to show that is not of row full rank. If it is not the case, let be the sparse difference resultant of the original system, we have that there exists an , such that . Then, the algebraic symbolic support matrix of is of full rank. Compare with , we find that the co-rank of is no more than which contradicts to the definition of .
3.2 A new algorithm for sparse difference resultant
Based on the new bounds and propositions for the sparse difference resultant, we propose an improved algorithm to compute sparse difference resultant for any given Laurent transformally essential difference polynomial system. The algorithm is motivated by the fact that sparse difference resultant is equal to the algebraic sparse resultant of a strong essential polynomial system which is derived from the original difference polynomial system. Thus one can transform the computation of sparse difference resultant to the computation of algebraic sparse resultant which has mature algorithms such as subdivision method initiated by Canny and Emiris .
Therefore, the algorithm is divided into two parts. The first part is to find the strong essential polynomial system. The main strategy for this part is to use the symbolic support matrix of the given difference polynomial system to determine the existence of sparse difference resultant and if yes, to obtain the unique super-essential system, and then simplify the super-essential system based on Theorem 3.3 and use algebraic tools to find the strong essential system. The second one is to use the mixed subdivision method to construct matrix representation of sparse resultant of the strong essential system whose determinant is the required sparse difference resultant up to a sign.
In order to present a better understanding of the whole procedure, we give the flow chart of the algorithm in Figure 1.
Figure 1. Flow chart of the algorithm
|Input:||A generic difference system .|
|Output:||The sparse difference resultant of .|
1. Construct the symbolic support matrix of .
If rank(, then proceed to compute SDResultant;
Else, return “No SDResultant for ”.
2. Set .
3. Let , choose an element .
3.2 If rank(, set , go back to step 3.
3.2.1 Else if , go to the next step,
3.2.2 else choose an go back to step 3.1.
Note is a super-essential system.
4. Assume . Compute the rank of the symbolic support matrix of by Gauss elimination. We obtain such that the matrix which corresponds to the -th, -th columns of has rank . We assume these columns are the first columns. Set the variables in to 1, we denoted by the new system under this substitution.
Compute the order matrix of and the common factor of the -th column of , compute the Jacobi number
Construct a new algebraic system .
5. Compute the algebraic symbolic support matrix of , let .
Find the algebraic essential system with minimal ranking from