Subresultant Sequence Research Articles

Lexicographic Gröbner bases of bivariate polynomials modulo a univariate one

Let T(x)∈k[x] be a monic non-constant polynomial and write R=k[x]/〈T〉 the quotient ring. Consider two bivariate polynomials a(x,y),b(x,y)∈R[y]. In a first part, T=pe is assumed to be the power of an irreducible polynomial p. A new algorithm that computes a minimal lexicographic Gröbner basis of the ideal 〈a,b,pe〉, is introduced. A second part extends this algorithm when T is general through the “local/global” principle realized by a generalization of “dynamic evaluation”, restricted so far to a polynomial T that is squarefree. The algorithm produces splittings according to the case distinction “invertible/nilpotent”, extending the usual “invertible/zero” in classic dynamic evaluation. This algorithm belongs to the Euclidean family, the core being a subresultant sequence of a and b modulo T. In particular no factorization or Gröbner basis computations are necessary. The theoretical background relies on Lazard's structural theorem for lexicographic Gröbner bases in two variables. An implementation is realized in Magma. Benchmarks show clearly the benefit, sometimes important, of this approach compared to the Gröbner bases approach.

A fraction-free unit-circle zero location test for a polynomial with any singularity profile

ABSTRACTThis paper presents a fraction-free (FF) version of the Bistritz test to determine the zero location (ZL) of a polynomial with respect to the unit circle. The test has the property that when it is invoked on a polynomial with Gaussian or real integer coefficients, it is an efficient integer algorithm completed without fractions over the respective integral domain. The test is not restricted to integers but remains integer preserving (IP) in all possible encounters of abnormalities and singularities. We define a symmetric subresultant polynomial sequence (SSPS) for the Sylvester matrix of two symmetric polynomials. We then show that the sequence of polynomials produced by the FF test coincides with the SSPS of its first two polynomials when the test is normal and the SSPS is strongly nonsingular, or else its polynomials match the non-singular subresultant polynomial and pass over intermediate gaps of singular subresultants in an IP and efficient manner. This relationship (interesting in its own right) is used to show that the test is IP and normally attains integers of minimal size.

New algebraic conditions for the identification of the relative position of two coplanar ellipses

The identification of the relative position of two real coplanar ellipses can be reduced to the identification of the nature of the singular conics in the pencil they define and, in general, their location with respect to these singular conics in the pencil. This latter problem reduces to find the relative location of the roots of univariate polynomials. Since it is usually desired that all generated expressions are algebraic to simplify further analysis, including the case in which the ellipses undergone temporal variations, all recent methods available in the literature rely mathematical tools such as Sturm–Habicht sequences or subresultant sequences. This paper presents an alternative based on more elementary tools which results in a binary decision tree to classify the relative location of two ellipses in 12 different classes. The decision at each node is taken based on the sign of a set of algebraic/rational expressions on the ellipses coefficients, the most complex of them being third and second order polynomial discriminants.

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.

Algebraic phase unwrapping along the real axis: extensions and stabilizations

The unwrapped phase of a complex function is defined with a line integral of the gradient of the arctangent of the ratio of the real and imaginary parts of the function. The phase unwrapping, which is a problem to reconstruct the unwrapped phase of an unknown complex function from its finite observed samples, has been a key for estimating useful physical quantity in many signal and image processing applications. In the light of the functional data analysis, it is natural to estimate first the unknown complex function by a certain piecewise complex polynomial and then to compute the exact unwrapped phase of the piecewise complex polynomial with the algebraic phase unwrapping algorithms (Yamada et al. in IEEE Trans Signal Process 46(6), 1639---1664, 1998; Yamada and Bose in IEEE Trans Circuits Syst I Fundam Theory Appl 49(3), 298---304, 2002; Yamada and Oguchi in Multidimens Syst Signal Process 22(1---3), 191---211, 2011). In this paper, we propose several useful extensions and numerical stabilizations of the algebraic phase unwrapping along the real axis which was established originally in Yamada and Oguchi (Multidimens Syst Signal Process 22(1---3), 191---211, 2011). The proposed extensions include (i) removal of a certain critical assumption premised in the original algebraic phase unwrapping, and (ii) algebraic phase unwrapping for a pair of bivariate polynomials. Moreover, in order to resolve certain numerical instabilities caused by the coefficient growth in an inductive step in the original algorithm, we propose to compute directly a certain subresultant sequence without passing through the inductive step. The extensive numerical experiments exemplify the notable improvement, in the performance of the algebraic phase unwrapping, made by the proposed numerical stabilization.

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.

An algebraic approach to continuous collision detection for ellipsoids

We present algebraic expressions for characterizing three configurations formed by two ellipsoids in R 3 that are relevant to collision detection: separation, external touching and overlapping. These conditions are given in terms of explicit formulae expressed by the subresultant sequence of the characteristic polynomial of the two ellipsoids and its derivative. For any two ellipsoids, the signs of these formulae can easily be evaluated to classify their configuration. Furthermore, based on these algebraic conditions, an efficient method is developed for continuous collision detection of two moving ellipsoids under arbitrary motions.

On the Topology of Real Algebraic Plane Curves

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition also induces a new approach for computing an arrangement of polylines isotopic to the input curve. We also present an analysis of the complexity of our algorithm. An implementation of our algorithm demonstrates its efficiency, in particular on high-degree non-generic curves.

Division-free computation of subresultants using Bezout matrices

We present an algorithm to compute the subresultant sequence of two polynomials that completely avoids division in the ground domain, generalizing an algorithm given by Abdeljaoued et al. [J. Abdeljaoued, G. Diaz-Toca, and L. Gonzalez-Vega, Minors of Bezout matrices, subresultants and the parameterization of the degree of the polynomial greatest common divisor, Int. J. Comput. Math. 81 (2004), pp. 1223–1238]. We evaluate determinants of slightly manipulated Bezout matrices using the algorithm of Berkowitz. Although the algorithm gives worse complexity bounds than pseudo-division approaches, our experiments show that our approach is superior for input polynomials with moderate degrees if the ground domain contains indeterminates.

Solving the implicitization, inversion and reparametrization problems for rational curves through subresultants

From the rational functions defining a rational plane curve it is possible to construct two bivariate polynomials that can be seen as univariate polynomials in the parameter value. In this paper several relevant properties of the subresultant sequence of these two polynomials are introduced which are used to solve simultaneously the implicitization, inversion, properness and reparametrization problems. It is also shown that these methods can be suitably transformed in order to deal with curves in the three-dimensional space presented by a polynomial parametrization.

Automatic computation of the complete root classification for a parametric polynomial

An improved algorithm, together with its implementation, is presented for the automatic computation of the complete root classification of a real parametric polynomial. The algorithm offers improved efficiency and a new test for non-realizable conditions. The improvement lies in the direct use of ‘sign lists’, obtained from the discriminant sequence, rather than ‘revised sign lists’. It is shown that the discriminant sequences, upon which the sign lists are based, are closely related both to Sturm–Habicht sequences and to subresultant sequences. Thus calculations based on any of these quantities are essentially equivalent. One particular application of complete root classifications is the determination of the conditions for the positive definiteness of a polynomial, and here the new algorithm is applied to a class of sparse polynomials. It is seen that the number of conditions for positive definiteness remains surprisingly small in these cases.

Bezout matrices, Subresultant polynomials and parameters

The main purpose of this paper is to analyze new determinantal expressions which define the subresultant sequence of two polynomials, when the coefficients of such polynomials depend on parameters.

On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of O ˜ B ( N 14 ) for the purely projection-based method, and O ˜ B ( N 12 ) for two subresultant-based methods: this notation ignores polylogarithmic factors, where N bounds the degree, and the bitsize of the polynomials. The previous record bound was O ˜ B ( N 14 ) . Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in O ˜ B ( N 12 ) , whereas the previous bound was O ˜ B ( N 14 ) . All algorithms have been implemented in maple, in conjunction with numeric filtering. We compare them against fgb/rs, system solvers from synaps, and maple libraries insulate and top , which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/ C++ libraries.

D-resultant and subresultants

We establish a connection between the D-resultant of two polynomials f(t) and g(t) and the subresultant sequence of f(t)-x and g(t)-y. This connection is used to decide in a more explicit way whether K(f(t), g(t)) = K(t) or K[f(t),g(t)] = K[t]. We also show how to extract a faithful parametrization from a given one.

Birational properties of the gap subresultant varieties

In this paper we address the problem of understanding the gaps that may occur in the subresultant sequence of two polynomials. We define the gap subresultant varieties and prove that they are rational and have the expected dimension. We also give explicitly their corresponding prime ideals.

Deciding Hopf Bifurcations by Quantifier Elimination in a Software-component Architecture

In this paper we give a semi-algebraic description of Hopf bifurcation fixed points for a given parameterized polynomial vector field. The description is carried out by use of the Hurwitz determinants, and produces a first-order formula which is transformed into a quantifier-free formula by the use of usual-quantifier elimination algorithms. We apply techniques from the theory of sub-resultant sequences and of Gröbner bases to come up with efficient reductions, which lead to quantifier elimination questions that can often be handled by existing quantifier elimination packages. We could implement the algorithms for the conditions on Hopf bifurcations by combining the computer algebra system Maple with packages for quantifier elimination using a Java-based component architecture recently developed by the second author. In addition to some textbook examples we applied our software system to an example discussed in a recent research paper.

Solving a congruence on a graded algebra by a subresultant sequence and its application

Let Γ be a submonoid of nonnegative integers. Let g be an element of Γ-graded algebra with a Γ-basis, and f be an element of a nonzero ideal of the algebra. In this paper, we consider solving for p and r the congruence pf  r (mod g) such that deg( r) - deg( p) is less than a given integer and p has the minimal degree among all solutions. We show how to solve the congruence by using a so-called subresultant sequence of f and g. We also give an algorithm to find a solution explicitly. As an application, we use this algorithm to decode geometric Goppa codes.

Decoding geometric Goppa codes using an extra place

Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the codelength is smaller than the number of rational points on the curve, then this method can correct up to 1.2 (d*-L)/2-s errors, where d* is the designed minimum distance of the code and s is the Clifford defect. The affine ring with respect to a place P is the set of all rational functions which have no poles except at P, and it is somehow similar to a polynomial ring. For a special kind of geometric Goppa code, namely C/sub Omega /(D,mP), the decoding algorithm is reduced to solving the key equation in the affine ring, which can be carried out by the subresultant sequence in the affine ring with complexity O(n/sup 3/), where n is the length of codewords.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>