The well-established theory of multivariate polynomial ideals (over C) was found in recent years to be important for the investigation of several problems in multivariate approximation. In this note we draw connections between the validity of spectral synthesis in the space of all multivariate sequences and questions in multivariate splines on uniform meshes. AMS (MOS) Subject Classifications: ...

The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M , and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte pol...

We show that rational data of bounded input length are uniformly distributed with respect to condition numbers of numerical analysis. We deal both with condition numbers of Linear Algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we show that for any w > 1 and for any n× n rational matrix M of bit length O(n4 log n)+logw, the condition number k(M...

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of Numerical Analysis. We deal both with condition numbers of Linear Algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n×n ...

Continuing our recent work in [5] we study polynomial masks of multivariate tight wavelet frames from two additional and complementary points of view: convexity and system theory. We consider such polynomial masks that are derived by means of the unitary extension principle from a single polynomial. We show that the set of such polynomials is convex and reveal its extremal points as polynomials...

We propose a novel approach to solve systems of multivariate polynomial equations, using the column space Macaulay matrix that is constructed from coefficients these polynomials. A multidimensional realization problem in null results an eigenvalue problem, eigenvalues and eigenvectors which yield common roots system. Since this based algorithm uses well-established numerical linear algebra tool...

The generating function that records the sizes of directed circuit partitions of a connected 2-in, 2-out digraph D can be determined from the interlacement graph of D with respect to a directed Euler circuit; the same is true of the generating functions for other kinds of circuit partitions. The interlace polynomials of Arratia, Bollobás and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; C...