The polynomial space H in the span of the integer translates of a box spline M admits a well-known characterization as the joint kernel of a set of homogeneous differential operators with constant coefficients. The dual space H∗ has a convenient representation by a polynomial space P, explicitly known, which plays an important role in box spline theory as well as in multivariate polynomial inte...

We describe a connection between discrete birth process and a certain family of multivariate interpolation polynomials. This enables us to compute all asymptotic moments of the birth process, generalizing previously known results for the mean and variance.

We exhibit sharp upper bounds for the probability distribution of the distance from a system of multivariate polynomial equations to the strata of all systems having a critical zero of given corank. We also prove sharp upper bounds for the probability distribution of the condition number of singular systems of multivariate polynomial equations. We finally state a new and sharp technique of the ...

The standard approach to factor a multivariate polynomial in Z[x1, x2, . . . , xn] is to factor a univariate image in Z[x1] then lift the factors of the image one variable at a time using Hensel lifting to recover the multivariate factors. At each step one must solve a multivariate polynomial Diophantine equation. For polynomials in many variables with many terms we find that solving these mult...

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 of f(β) for a sufficiently large number β. In the multivariate case, we introduce the modified Kronecker substitution to reduce the interpolation of a multivariate polynomial to the univariate case. Both algorith...

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. F...

Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Féjer-Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. F...

We define a multivariate polynomial that generalizes in a unified way the twovariable interlace polynomial defined by Arratia, Bollobás and Sorkin on the one hand, and a one-variable variant of it defined by Aigner and van der Holst on the other. We determine a recursive definition for our polynomial that is based on local complementation and pivoting like the recursive definitions of Tutte’s p...

Masry (1996b) provides estimation bias and variance expression for a general local polynomial kernel estimator in a general multivariate regression framework. Under smoother conditions on the unknown regression and by including more refined approximation terms than that in Masry (1996b), we extend the result of Masry (1996b) to obtain explicit leading bias terms for the whole vector of the loca...

When the number of objects in the training set is too small for the number of features used, most classification procedures cannot find good classification boundaries. In this paper, we introduce a new technique to solve the one class classification problem based on fitting an implicit polynomial surface to the point cloud of features to model the one class which we are trying to separate from ...