Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of a sum of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We disscuss effective methods to get a simpler representation of a “sparse” polynomial as a sum of squares of sparse pol...

A new method for inversion of rectangular matrices in a multivariate polynomial ring with coefficients in a field is explained. This method requires that the polynomial matrix satisfies the one-to-one mapping criteria defined in [1].

QUAD stream cipher uses multivariate polynomial systems. It has provable security based on the computational hardness assumption. More specifically, the security of QUAD depends on hardness of solving non-linear multivariate systems over a finite field, and it is known as an NP-complete problem. However, QUAD is slower than other stream ciphers, and an efficient implementation, which has a redu...

Representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and SDP (semidefinite programming) relaxation of polynomial optimization problems. We disscuss effective methods to obtain a simpler representation of a “sparse” polynomial as a sum of squares of sparse p...

We use the Krammer representation of the braid group in Libgober’s invariant and construct a new multivariate polynomial invariant for curve complements: Krammer polynomial. We show that the Krammer polynomial of an essential braid is equal to zero. We also compute the Krammer polynomials of some certain n-gonal curves.

We present two results, the first on the distribution of the roots of a polynomial over the ring of integers modulo n and the second on the distribution of the roots of the Sylvester resultant of two multivariate polynomials. The second result has application to polynomial GCD computation and solving polynomial diophantine equations.

A straightforward way to represent multivariate polynomial in software is to implement them recursively as univariate polynomials over a polynomial ring. This is especially common in an object oriented context. We present a short algorithm which maps polynomials from one polynomial ring to another polynomial ring where the order of variables is permuted. This algorithm uses the recursive repres...

Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. We show that within this representation the polynomial greatest common divisor and factorization problems, as well as the problem of extracting the numerator and denominator of a rational function, can all b...