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About 2,270 papers for “multivariate polynomials”

Multivariate Polynomials

Manfred Reimer · Birkhäuser Basel eBooks · 2003

The theory of multivariate polynomial approximation is characterized by a great variety of polynomials which can be used, and also by a great richness of geometric situations which occur. This chapter presents the most important facts on multivariate polynomials.

Cited by 1

Multivariate Polynomials

Peter J. Olver · Cambridge University Press eBooks · 1999

We have now reached the culmination of our brief tour of the most noteworthy features in the invariant theory of binary forms and stand ready to launch into new and less well charted territories. Unfortunately, our space limitations prevent us from realizing this grander aim in any depth, but we do have room in this final chapter for a brief introduction to the invariant theory of homogeneous polynomials in more than two variables (along with their inhomogeneous counterparts). Not surprisingly, almost all of the classical — and not so classical — methods that have been developed for binary forms can be straightforwardly adapted to the multivariate situation. On the other hand, the required computations are considerably more involved, and much less is known about explicit bases for invariants and covariants, solution to equivalence problems, canonical forms, structure of symmetries, and so on.

Cited by 0

Multivariate orthogonal polynomials

Leonard W. Cooper · VTechWorks (Virginia Tech) · 1951

The object of this thesis is to define special orthogonal polynomials and develop efficient methods for employing them which have the same advantages with respect to functions of the type (1.2) as do univariate orthogonal polynomials in the simple case k=1. These new polynomials may be usefully termed “multivariate orthogonal polynomials.”

Cited by 3

Pseudozeros Of Multivariate Polynomials

J. William Hoffman, James J. Madden, HONG ZHANG · 2000

. The pseudozero set of a system f of polynomials in n complex variables is the subset of C n which is the union of the zero - sets of all polynomial systems g that are near to f in a suitable sense. This concept is made precise and general properties of pseudozero sets are established. In particular it is shown that under wide circumstances, the pseudozero set is a semialgebraic set. Also, estimates are given for the size of the projections of pseudozero sets into coordinate directions. Several examples are presented illustrating some of the general theory developed here. Finally, algorithmic ideas are proposed for solving multivariate polynomials. 1. Introduction 1.1. Summary. The pseudozero set of a general polynomial in a single variable was investigated in [18]. Our purpose here is to extend some ideas from that work to systems of polynomials in several variables, with special attention to the case in which the zero set of the system consists of finitely many points. Recall th...

Cited by 0

Pseudozeros of multivariate polynomials

William Hoffman, J., Madden, James J., Zhang, Hong · Civil War Book Review · 2003

The pseudozero set of a system f of polynomials in n complex variables is the subset of Cn which is the union of the zero-sets of all polynomial systems g that are near to f in a suitable sense. This concept is made precise, and general properties of pseudozero sets are established. In particular it is shown that in many cases of natural interest, the pseudozero set is a semialgebraic set. Also, estimates are given for the size of the projections of pseudozero sets in coordinate directions. Several examples are presented illustrating some of the general theory developed here. Finally, algorithmic ideas are proposed for solving multivariate polynomials.

Cited by 0

On multivariate minimal polynomials

Thomas Bloom, Jean-Paul Calvi · Mathematical Proceedings of the Cambridge Philosophical Society · 2000

Given compact sets E and F in [Copf ]n (n [ges ] 1) related by F = q−1(E) where q is a polynomial map, we are interested in the general problem of comparing minimal polynomials for E with minimal polynomials for F. Let α be an n-multi-index of length d. We define the classes of polynomials ℙ(α) := zα + [Copf ]d−1[z] and [Pscr ](α) := {p: p(z) = zα + [sum ]β[pr ]αaβzβ} where [pr ] denotes the usual graded lexicographic order. Polynomials in ℙ(α) or in [Pscr ](α) of least deviation from zero on E (with respect to the supremum norm) are called minimal polynomials for E. We prove that if q is a simple (i.e. qˆi(z) = zmi) polynomial mapping of degree m and if p is minimal polynomial for E then p ∘ q is a minimal polynomial for F = q−1(E) and, using some algebraic machinery, we can also construct minimal polynomials for E from minimal polynomials for F. The result seems to be new even in the one-dimensional case.

Cited by 14

Multivariate circular Jacobi polynomials

Genki Shibukawa · arXiv (Cornell University) · 2014

We introduce a new multivariate orthogonal polynomial which is a 2-parameter deformation of the spherical polynomial by harmonic analysis on symmetric cone. This is also regarded as a multivariate analogue of the circular Jacobi polynomial. Further, the weight function of its orthogonality relation coincides with the circular Jacobi ensemble defined by Bourgade et al.. We also obtain its main properties : generating function, pseudo differential equation and deteraminant expression.

Cited by 0Open access

Factoring larger multivariate polynomials

Paul S. Wang · ACM SIGSAM Bulletin · 1976

An improved algorithm for factoring multivariate polynomials over the integers has been developed. For larger polynomials, it is generally faster and requires less storage than the original algorithm as described by Wang and Rothschild [2]. The new algorithm has improved strategy to deal with the known problems of the original algorithm. namely, the leading coefficient problem, the bad-zero problem and the combinatorial search for true factors. It features a linearly convergent variable-by-variable p-adic construction procedure for parallel lifting of all factors at once. This procedure is very efficient and it relies on i) distributing the factors of the leading coefficient correctly, and ii) solving the equation αf+βg = h, with multivariate f, g and h, efficiently. Details of these and other aspects of this algorithm together with its generalization to factoring over algebraic number fields will be forthcoming. Obviously, the EZ-GCD algorithm [1] can be similarly improved if its lifting procedure is replaced.

Cited by 0

On Multivariate Orthogonal Polynomials

Yuan Xu · SIAM Journal on Mathematical Analysis · 1993

Orthogonal polynomials in several variables are studied. The results include a newformulation of the recurrence relation, characterization of orthogonality of polynomial sequences, an analogy of Christoffel-Darboux formula, and properties of reproducing kernel function.

Cited by 96

Multivariate Polynomials in Sage

Viviane Pons · arXiv (Cornell University) · 2012

We have developed a patch implementing multivariate polynomials seen as a multi-base algebra. The patch is to be released into the software Sage and can already be found within the Sage-Combinat distribution. One can use our patch to define a polynomial in a set of indexed variables and expand it into a linear basis of the multivariate polynomials. So far, we have the Schubert polynomials, the Key polynomials of types A, B, C, or D, the Grothendieck polynomials and the non-symmetric Macdonald polynomials. One can also use a double set of variables and work with specific double-linear bases like the double Schubert polynomials or double Grothendieck polynomials. Our implementation is based on a definition of the basis using divided difference operators and one can also define new bases using these operators.

Cited by 2Open access