Polynomial Remainder Sequence Research Articles

A Bridge between Euclid and Buchberger: (An Attempt to Enhance Gröbner Basis Algorithm by PRSs and GCDs)

This article surveys a very new method of enhancing Buchberger's Gröbner basis algorithm by the PRSs (polynomial remainder sequences) and the GCDs of multivariate polynomials. Let F = { F 1 ,..., F m +1 } ⊂ K[ x , u ] be a given system, where ( x ) = ( x 1 ,..., x m ) and ( u ) = ( u 1 ,..., u n ). Currently, we treat only such F s that are "healthy" (see the text). Let [EQUATION], where [EQUATION], be the reduced Gröbner basis of ideal ( F ) w.r.t. the lexicographic order, to be abbreviated to GB( F ). Let [EQUATION], be such that [EQUATION] is a small multiple of G 1 , and the leading monomial of [EQUATION], is a multiple (hopefully small) of the leading monomial of G i . Our method computes [EQUATION] first, then computes [EQUATION]. Finally, we will apply Buchberger's method to system [EQUATION]. Four new theorems are given. The first and second ones are to compute the lowest-order element of the ideal generated by relatively prime G , H ∈ K[ x , u ], without computing any Spolynomial. The third theorem says that if F is healthy then [EQUATION]. We compute resultants in K[ u ], of F through different routes. Then, by Theorem 3, the resultants will be different multiples of G 1 . Hence, the GCD of them will be a small multiple of G 1 . In the elimination of x through different routes, we obtain sets of similar remainders such that the elements of each set have the same leading variable and nearly the same degrees. We call the leading coefficients of mutually similar remainders an "LCsystem". We eliminate the leading variables of suitably chosen LCsystems. The fourth theorem constructs a polynomial [EQUATION], such that the leading coefficient of [EQUATION] is the GCD of resultants of elements of an LCsystem chosen.

Subresultant Polynomial Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x

In this paper we present two new methods for computing thesubresultant polynomial remainder sequence (prs) of two polynomials f, g ∈ Z[x]. We are now able to also correctly compute the Euclidean and modifiedEuclidean prs of f, g by using either of the functions employed by ourmethods to compute the remainder polynomials.Another innovation is that we are able to obtain subresultant prs’s inZ[x] by employing the function rem(f, g, x) to compute the remainderpolynomials in [x]. This is achieved by our method subresultants_amv_q(f, g, x), which is somewhat slow due to the inherent higher cost of com-putations in the field of rationals.To improve in speed, our second method, subresultants_amv(f, g,x), computes the remainder polynomials in the ring Z[x] by employing thefunction rem_z(f, g, x); the time complexity and performance of thismethod are very competitive.Our methods are two different implementations of Theorem 1 (Section 3),which establishes a one-to-one correspondence between the Euclidean andmodified Euclidean prs of f, g, on one hand, and the subresultant prs of f, g,on the other.By contrast, if – as is currently the practice – the remainder polynomi-als are obtained by the pseudo-remainders function prem(f, g, x) 3 , thenonly subresultant prs’s are correctly computed. Euclidean and modified Eu-clidean prs’s generated by this function may cause confusion with the signsand conflict with Theorem 1.ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.

A Basic Result on the Theory of Subresultants

Given the polynomials f, g ∈ Z[x] the main result of our paper,Theorem 1, establishes a direct one-to-one correspondence between the modified Euclidean and Euclidean polynomial remainder sequences (prs’s) of f, gcomputed in Q[x], on one hand, and the subresultant prs of f, g computedby determinant evaluations in Z[x], on the other.An important consequence of our theorem is that the signs of Euclideanand modified Euclidean prs’s - computed either in Q[x] or in Z[x] - are uniquely determined by the corresponding signs of the subresultant prs’s. In this respect, all prs’s are uniquely "signed".Our result fills a gap in the theory of subresultant prs’s. In order to placeTheorem 1 into its correct historical perspective we present a brief historicalreview of the subject and hint at certain aspects that need - according toour opinion - to be revised.ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.

Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences

In 1971 using pseudo-divisions - that is, by working in Z[x] - Brown and Traub computed Euclid’s polynomial remainder sequences (prs’s) and (proper) subresultant prs’s using sylvester1, the most widely known form of Sylvester’s matrix, whose determinant defines the resultant of two polynomials. In this paper we use, for the first time in the literature, the Pell-Gordon Theorem of 1917, and sylvester2, a little known form of Sylvester’s matrix of 1853 to initially compute Sturm sequences in Z[x] without pseudodivisions - that is, by working in Q[x]. We then extend our work in Q[x] and, despite the fact that the absolute value of the determinant of sylvester2 equals the absolute value of the resultant, we construct modified subresultant prs’s, which may differ from the proper ones only in sign.

Multiplicity-preserving triangular set decomposition of two polynomials

In this paper, a multiplicity-preserving triangular set decomposition algorithm is proposed for a system of two polynomials, which involves only computing the primitive polynomial remainder sequence of two polynomials once and certain GCD computations. The algorithm decomposes the unmixed variety defined by two polynomials into square free and disjoint (for non-vertical components, see Definition 4) algebraic cycles represented by triangular sets, which may have negative multiplicities. Thus, the authors can count the multiplicities of the non-vertical components. In the bivariate case, the authors give a complete algorithm to decompose the system into zeros represented by triangular sets with multiplicities. The authors also analyze the complexity of the algorithm in the bivariate case. The authors implement the algorithm and show the effectiveness of the method with extensive

Improved polynomial remainder sequences for Ore polynomials

Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.

Computing the polynomial remainder sequence via Bézout matrices

In this paper, we address the task of computing the polynomial remainder sequence appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, m<n via renewed approach based on the approximate block diagonalization of a Bézout matrix B(u,v). More specifically, this algorithm consists of using the efficient Schur complementation method given in terms of a Bézout matrix B(u,v) associated with the input polynomials. We also compare the proposed approach to the approximate block diagonalization of a Hankel matrix H(u,v) in which the successive transformation matrices are upper triangular Toeplitz matrices. All algorithms have been implemented in Matlab. Numerical experiments performed with a wide variety of test problems show the effectiveness of this algorithm in terms of efficiency, stability and robustness.

Improved polynomial remainder sequences for ore polynomials

Polynomial remainder sequences contain the intermediate results of the Euclidean algorithm when applied to (non-)commutative polynomials. The running time of the algorithm is dependent on the size of the coefficients of the remainders. Different ways have been studied to make these as small as possible. The subresultant sequence of two polynomials is a polynomial remainder sequence in which the size of the coefficients is optimal in the generic case, but when taking the input from applications, the coefficients are often larger than necessary. We generalize two improvements of the subresultant sequence to Ore polynomials and derive a new bound for the minimal coefficient size. Our approach also yields a new proof for the results in the commutative case, providing a new point of view on the origin of the extraneous factors of the coefficients.

Properties of regular systems and algorithmic improvements for regular decomposition

In this paper, we study the properties of regular systems and improve the efficiency of the regular decomposition method RegSer implemented in Epsilon. We define a weaker concept which retains most properties of regular system. It can be shown that from a weak regular system one can also define a regular set and vice versa. We present an algorithm RecurWeakRegSer to decompose a given polynomial system [ P , Q ] into weak regular systems. When Q ≠ 0̸ , the output of RecurWeakRegSer( [ P , Q ] ) often contains fewer components than that of RegSer( [ P , Q ] ). This is one advantage of RecurWeakRegSer. Another one is that RecurWeakRegSer is more efficient than RegSer. This was shown by experiments that we carried out. Since it is an essential step in RegSer to compute subresultant polynomial remainder sequences (PRS), and there is some weakness in the implementation, we implement a new version of subresultant algorithm using the optimization strategy of Ducos so that the efficiency of RegSer can be improved.

Recursive polynomial remainder sequence and its subresultants

We introduce concepts of “recursive polynomial remainder sequence (PRS)” and “recursive subresultant,” along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated “recursively” for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is built-up just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a “nested” expression, i.e. a Sylvester matrix whose elements are themselves determinants.

Subresultants revisited

Subresultants and polynomial remainder sequences are an important tool in polynomial computer algebra. In this survey, we sketch the history, formalize a unified framework for the various notions, derive a number of results from the early 1970s within our framework, and report on implementations.

Polynomial remainder sequence and approximate GCD

Let P 1 and P 2 be polynomials, univariate or multivariate, and let (P 1 , P 2 , P 3 ,&amp;hellip;, P i ,&amp;hellip;) be a polynomial remainder sequence. Let A i and B i (i = 3, 4,&amp;hellip;) be polynomials such that A i P 1 + B i P 2 = P i , deg(A i ) &amp;lt; deg(P 2 ) - deg(P i ), deg(B i ) &amp;lt; deg(P 1 ) - deg(P i ), where the degree is for the main variable. We derive relations such as C i P 1 = -B i+1 P i + B i P i+1 and C i P 2 = A i+1 P i - A i P i+1 , where C i is independent of the main variable. Using these relations, we discuss approximate common divisors calculated by polynomial remainder sequence.

A Stable Extended Algorithm for Generating Polynomial Remainder Sequence

A Stable Extended Algorithm for Generating Polynomial Remainder Sequence

Various proofs of Sylvester's (determinant) identity

Despite the fact that the importance of Sylvester's determinant identity has been recognized in the past, we were able to find only one proof of it in English (Bareiss, 1968), with reference to some others. (Recall that Sylvester (1857) stated this theorem without proof.) Having used this identity, recently, in the validity proof of our new, improved, matrix-triangularization subresultant polynomial remainder sequence method (Akritas et al., 1995), we decided to collect all the proofs we found of this identity-one in English, four in German and two in Russian, in that order-in a single paper (Akritas et al., 1992). It turns out that the proof in English is identical to an earlier one in German. Due to space limitations two proofs are omitted.

Matrix computation of subresultant polynomial remainder sequences in integral domains

We present an improved variant of the matrix-triangularization subresultant prs method [1] for the computation of a greatest common divisor of two polynomialsA andB (of degreesm andn, respectively) along with their polynomial remainder sequence. It is improved in the sense that we obtain complete theoretical results, independent of Van Vleck’s theorem [13] (which is not always true [2, 6]), and, instead of transforming a matrix of order 2·max(m, n) [1], we are now transforming a matrix of orderm+n. An example is also included to clarify the concepts.

The Berlekamp-Massey algorithm and linear recurring sequences over a factorial domain

We present an extended polynomial remainder sequence algorithm XPRS for R[X] whereR is a domain. From this we derive a Berlekamp-Massey algorithm BM/R overR. We show that if (α) is a linear recurring sequence in a factorial domainU, then the characteristic polynomials for (α) form aprincipal ideal which is generated by a primitive minimal polynomial. Moreover, this generator ismonic when U[[X]] is factorial (for example, whenU is Z orK[X1,X2,...,Xn] whereK is a field). From XPRS we derive an algorithm MINPOL for determining the minimal polynomial of (α) when an upper bound on the degree of some characteristic polynomial and sufficiently many initial terms of (α) are known. We also show how to obtain a Berlekamp-Massey type minimal polynomial algorithm from BM/U and state BM_MINPOL/K explicitly with a further refinement. Examples are given forU=Z, GF(2)[Y].

Primitive polynomial remainder sequences in elimination theory

Primitive polynomial remainder sequences (pprs) are more than a tool for computing gcd's; the content computations in the course of computing the pprs of two multivariate polynomialsf 1 andf 2 provide information on the common zeros off 1 andf 2. Because of this additional property, primitive polynomial remainder sequences can be used for solving systems of algebraic equations.

Fast Parallel Computation of the Polynomial Remainder Sequence via Bézout and Hankel Matrices

If $u(x)$ and $v(x)$ are polynomials of degree n and m, respectively, $m < n$, all the coefficients of the polynomials generated by the Euclidean scheme applied to $u(x)$ and $v(x)$ can be computed by using $O(\log^{3} n)$ parallel arithmetic steps and $n^{2}/ \log n$ processors over any field of characteristic 0 supporting FFT (Fast Fourier Transform). If the field does not support FFT the number of processors is increased by a factor of $\log \log n$; if the field does not allow division by $n!$ the number of processors is increased by a factor of n. This result is obtained by reducing the Euclidean scheme to computing the block triangular factorization of the Bezout matrix associated with $u(x)$ and $v(x)$. This approach is also extended to the evaluation of polynomial gcd (greatest common divisor) over any field of constants in $O(\log^{2} n)$ steps with the same number of processors.

Improvements of the Power-series Coefficient Polynomial Remainder Sequence GCD algorithm

The PC-PRS (Power-series Coefficient Polynomial Remainder Sequence) GCD is a multivariate polynomial GCD algorithm constructed recently, and is quite practical. The algorithm calculates a PRS by treating coefficients w.r.t. the main variable as a truncated power-series. Efficiency of the PC-PRS GCD algorithm depends greatly on how the power-series is truncated, and the algorithm employs a degree bound of GCD, determined theoretically. This theoretical bound is often larger than the true bound of GCD. In the first part of this paper, a practical bound is introduced so as to improve the computing time of the original algorithm. The practical bound is, however, not always large enough to calculate the true GCD. When the practical bound is small, the construction methods are applied to lift the truncated polynomials to the true GCD. In the remaining parts, the improved algorithm is combined with three construction methods; one is the Hensel construction, and the other two use the extended Euclidean relation and the division relation. Empirical tests show that our algorithms are efficient for polynomials of low to middle degree and many variables.

Three new algorithms for multivariate polynomial GCD

Three new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The first is to calculate a Gröbner basis with a certain term ordering. The second is to calculate the subresultant by treating the coefficients w.r.t. the main variable as truncated power series. The third is to calculate a PRS (polynomial remainder sequence) by treating the coefficients as truncated power series. The first algorithm is not important practically, but the second and third ones are efficient and seem to be useful practically. The third algorithm has been implemented naively and compared with the trial-division PRS algorithm and the EZGCD algorithm. Although it is too early to derive a definite conclusion, the PRS method with power series coefficients is very efficient for calculating low degree GCD of high degree non-sparse polynomials.