Computing with polynomials given byblack boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators

Given a system of polynomial equations with parameters, we present a new algorithm for computing its Dixon resultant R. Our algorithm interpolates the monic square-free factors of R one at a time from monic univariate polynomial images of R using sparse rational function interpolation. In this work, we use a modified version of the sparse multivariate rational function interpolation algorithm of Cuyt and Lee.We have implemented our new Dixon resultant algorithm in Maple with some subroutines coded in C for efficiency. We present timing results comparing our new Dixon resultant algorithm with Zippel’s algorithm for interpolating R and a Maple implementation of the Gentleman & Johnson minor expansion algorithm for computing R.KeywordsDixon resultantParametric polynomial systemsResultantSparse rational function interpolationKronecker substitution

Error-correcting decoding is generalized to multivariate sparse rational function recovery from evaluations that can be numerically inaccurate and where several evaluations can have severe errors (outliers). The generalization of the Berlekamp-Welch decoder to exact Cauchy interpolation of univariate rational functions from values with faults is by Kaltofen and Pernet in 2012. We give a different univariate solution based on structured linear algebra that yields a stable decoder with floating point arithmetic. Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D. Thesis MIT 1979] with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc. SNC 2007], and removes outliers (cleans up data) through techniques from error correcting codes. Our multivariate algorithm can build a sparse model from a number of evaluations that is linear in the sparsity of the model.

This paper describes the implementation of recursive algorithms in unique factorization domains, namely multivariate polynomial greatest common divisors (gcd) and factorization into irreducible parts in the Java computer algebra library (JAS). The implementation of gcds, resultants and factorization is part of the essential building blocks for any computation in algebraic geometry, in particular in automated deduction in geometry. There are various implementations of these algorithms in procedural programming languages. Our aim is an implementation in a modern object oriented programming language with generic data types, as it is provided by Java programming language. We exemplify that the type design and implementation of JAS is suitable for the implementation of several greatest common divisor algorithms and factorization of multivariate polynomials. Due to the design we can employ this package in very general settings not commonly seen in other computer algebra systems. As for example, in the coefficient arithmetic for advanced Grobner basis computations like in polynomial rings over rational function fields or (finite, commutative) regular rings. The new package provides factory methods for the selection of one of the several implementations for non experts. Further we introduce a parallel proxy for gcd implementations which runs different implementations concurrently.

We consider the problem of sparse interpolation of a multivariate black-box polynomial in floating-point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all values are represented in standard, fixed-precision, floating-point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give an efficient and numerically robust solution with high probability. We outline the numerical stability of our algorithm, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques through numerical experiments.

If the system of linear equations defining a multivariate rational interpolant is singular, then the table of multivariate rational interpolants displays a structure where the basic building block is a hexagon. Remember that for univariate rational interpolation the structure is built by joining squares. In this paper we associate with every entry of the table of rational interpolants a well-defined determinant representation, also when this entry has a nonunique solution. These determinant formulas are crucial if one wants to develop a recursive computation scheme. In section 2 we repeat the determinant representation for nondegenerate solutions (nonsingular systems of interpolation conditions). In theorem 1 this is generalized to an isolated hexagon in the table. In theorem 2 the existence of such a determinant formula is proven for each entry in the table. We conclude with an example in section 5.

Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We present a condition that guarantees that the equivariant Hilbert series is a rational function in m+1 variables with rational coefficients. Furthermore we give explicit formulas for the rational functions with coefficients in a number field and an algorithm for determining the rational functions with rational coefficients. A key is to construct finite automata that recognize languages corresponding to invariant filtrations.

We provide a simple method, using Grobner bases over modules, to compute multivariate polynomial greatest common divisors. The approach we show is flexible, adaptable to algebraic extensions of the rationals or prime fields, and is notably faster than prior methods that work with Grobner bases. It can be used in situations where sparse interpolation might be difficult to implement, e.g. when there are few points for interpolation (small prime fields) or in the presence of non-numeric algebraic relations.

Rational Functions are used to describe the self-excited forces acting on the bridge deck in the time domain. They can be identified indirectly based on aerodynamic derivatives or directly with the free (E2RFC method) or forced vibration technique, which can significantly decrease the testing time. The approach presented herein enables the extraction of Rational Function Coefficients by testing the section model at only one wind speed. This aim is achieved by increased complexity of the forced motion compared to the previous tests, which made it possible to test a wider range of reduced velocities by adjusting the motion frequency. In this study, motion histories generated from the assumed flat spectra are used. Wind tunnel tests on a streamlined section model utilizing simultaneous vertical, horizontal and torsional vibrations were performed to extract Rational Function Coefficients associated with 3-degree-of-freedom motion. Restrictions and improvements arising from the proposed methodology are described.

In this paper, we give new sparse interpolation algorithms for black box polynomial f whose coefficients are from a finite set. In the univariate case, we recover f from one evaluation \(f(\beta )\) for a sufficiently large number \(\beta \). In the multivariate case, we introduce the modified Kronecker substitution to reduce the interpolation of a multivariate polynomial to that of the univariate case. Both algorithms have polynomial bit-size complexity.

We show that the following are equivalent: (i) A rectangle of eccentricityv can be tiled using rectangles of eccentricityu. (ii) There is a rational function with rational coefficients,Q(z), such thatv =Q(u) andQ maps each of the half-planes {z ¦ Re(z) < 0} and {z ¦ Re(z) > 0 into itself, (iii) There is an odd rational function with rational coefficients,Q(z), such thatv = Q(u) and all roots ofv = Q(z) have a positive real part. All rectangles in this article have sides parallel to the coordinate axes and all tilings are finite. We letR(x, y) denote a rectangle with basex and heighty. In 1903 Dehn [1 ] proved his famous result thatR(x, y) can be tiled by squares if and only ify/x is a rational number. Dehn actually proved the following result. (See [4] for a generalization to tilings using triangles.)

Polynomial matrices play an important part in linear system calculations. New computational procedures are given for calculation of the Smith normal form and the greatest common right divisor of polynomial matrices. It is shown how suitable transformation matrices can be determined for the calculation of the Smith normal form, and how a set of polynomial matrix multipliers can be calculated for the greatest common right divisor problem. Neither of these algorithms relies on explicit calculation of the greate3t common divisor of polynomials. Limited numerical experience has shown that the3e algorithms are both fast and accurate.

Reconstructing rational functions with FireFly

We explore a special case of rational identity testing and algorithmic versions of two theorems on noncommutative polynomials, namely, Amitsur's theorem [S.A Amitsur, 1966] and the Bresar-Klep theorem [Bresar and Klep, 2008] when the input polynomial is given by an algebraic branching program (ABP). Let f be a degree-d n-variate noncommutative polynomial in the free ring Q over rationals. 1) We consider the following special case of rational identity testing: Given a noncommutative ABP as white-box, whose edge labels are linear forms or inverses of linear forms, we show a deterministic polynomial-time algorithm to decide if the rational function computed by it is equivalent to zero in the free skew field Q . Given black-box access to the ABP, we give a deterministic quasi-polynomial time algorithm for this problem. 2) Amitsur's theorem implies that if a noncommutative polynomial f is nonzero on k x k matrices then, in fact, f(M_1,M_2,...,M_n) is invertible for some matrix tuple (M_1,M_2,...,M_n) in (M_k(ℚ))^n. While a randomized polynomial time algorithm to find such (M_1,M_2,...,M_n) given black-box access to f is simple, we obtain a deterministic s^{O(log d)} time algorithm for the problem with black-box access to f, where s is the minimum ABP size for f and d is the degree of f. 3) The Bresar-Klep Theorem states that the span of the range of any noncommutative polynomial f on k x k matrices over Q is one of the following: zero, scalar multiples of I_k, trace-zero matrices in M_k(Q), or all of M_k(Q). We obtain a deterministic polynomial-time algorithm to decide which case occurs, given white-box access to an ABP for f. We also give a deterministic s^{O(log d)} time algorithm given black-box access to an ABP of size s for f. Our algorithms work when k >= d. Our techniques are based on some automata theory combined with known techniques for noncommutative ABP identity testing [Ran Raz and Amir Shpilka, 2005; Michael A. Forbes and Amir Shpilka, 2013].

Algebraic operations on polynomials and rational functions. Factors of polynomials. The remainder theorem. The factor theorem. Positive and negative rational indices.

We discuss the algorithms which, given a linear difference equation with rational function coefficients over a field k of characteristic 0, compute a polynomial U(x) ∈ k[x] (a universal denominator) such that the denominator of each of rational solutions (if exist) of the given equation divides U(x). We consider two types of such algorithms. One of them is based on constructing a set of irreducible polynomials that are candidates for divisors of denominators of rational solutions, and on finding a bound for the exponent of each of these candidates (the full factorization of polynomials is used). The second one is related to earlier algorithms for finding universal denominators, where the computation of gcd was used instead of the full factorization. The algorithms are applicable to scalar equations of arbitrary orders as well as to systems of first-order equations.A complexity analysis and a time comparison of the algorithms implemented in Maple are presented.KeywordsComputer AlgebraRational SolutionIrreducible PolynomialPolynomial SolutionPolynomial VersusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.