Tractability of multivariate problems has become nowadays a popular research subject. Polynomial tractability means that the solution of a d-variate problem can be solved to within ε with polynomial cost in ε−1 and d. Unfortunately, many multivariate problems are not polynomially tractable. This holds for all non-trivial unweighted linear tensor product problems. By an unweighted problem we mea...

Though Multivariate Polynomial Cryptography (MPC) is a potential candidate for Post Quantum Cryptography (PQC), schemes based on it are mostly well developed for digital signatures only. In this paper we explore the possibility of using multivariate polynomial systems for symmetric encryption and secret sharing.

The map H → H↓ assigns to each finite-dimensional space of smooth functions a homogeneous polynomial space of the same dimension. We discuss applications of this map in the areas of multivariate polynomial interpolation, box spline theory and polynomial ideals. §

The Cayley-Dixon formulation for multivariate projection operators (multiples of resultants of multivariate polynomials) has been shown to be efficient (both experimentally and theoretically) for simultaneously eliminating many variables from a polynomial system. In this paper, the behavior of the Cayley-Dixon projection operator and the structure of Dixon matrices are analyzed for composed pol...

In this paper, we apply the theory of multivariate polynomial matrices to the study of syzygy modules for a system of homogeneous linear equations with multivariate polynomial coefficients. Several interesting structural properties of syzygy modules are presented and illustrated with examples. © 1999 Elsevier Science Inc. All rights reserved. AMS classification: 15A23; 13P05; 15A24; 15A06; 13C9...

In this paper, we propose an approach to automatically compute invariant clusters for semialgebraic hybrid systems. An invariant cluster for an ordinary differential equation (ODE) is a multivariate polynomial invariant g(~u, ~x) = 0, parametric in ~u, which can yield an infinite number of concrete invariants by assigning different values to ~u so that every trajectory of the system can be over...

We study various versions of the univariate and multivariate matching and rook polynomials. We show that there is most general multivariate matching polynomial, which is, up the some simple substitutions and multiplication with a prefactor, the original multivariate matching polynomial introduced by C. Heilmann and E. Lieb. We follow here a line of investigation which was very successfully purs...

1 SMP-based parallel algorithms and implementations for polynomial factoring and GCD are overviewed. Topics include polynomial factoring modulo small primes, univariate and multivariate p-adic lifting, and reformulation of lift basis. Sparse polynomial GCD is also covered.

In this paper, we present a new method for computing boundeddegree factors of lacunary multivariate polynomials. In particular for polynomials over number fields, we give a new algorithm that takes as input a multivariate polynomial f in lacunary representation and a degree bound d and computes the irreducible factors of degree at most d of f in time polynomial in the lacunary size of f and in ...