Here we give a simple proof of a new representation for orthogonal polynomials over triangular domains which overcomes the need to make symmetry destroying choices to obtain an orthogonal basis for polynomials of fixed degree by employing redundancy. A formula valid for simplices with Jacobi weights is given, and we exhibit its symmetries by using the Bernstein–Bézier form. From it we obtain th...

The convergence properties of q-Bernstein polynomials are investigated. When q > 1 is fixed the generalized Bernstein polynomials Bnf of f , a one parameter family of Bernstein polynomials, converge to f as n → ∞ if f is a polynomial. It is proved that, if the parameter 0 < q < 1 is fixed, then Bnf → f if and only if f is linear. The iterates of Bnf are also considered. It is shown that B n f c...

This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in the case of one function. We also make an extensive study of an example for which we give an expression of a generic (and under some conditions, a relative) B...

It is well known that the degree-raised Bernstein-Bézier coefficients of degree n of a polynomial g converge to g at the rate 1/n. In this paper we consider the polynomial An(g) of degree ≤ n interpolating the coefficients. We show how An can be viewed as an inverse to the Bernstein operator and that the derivatives An(g) (r) converge uniformly to g at the rate 1/n for all r. We also give an as...

We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for l...

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety, using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration of multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein-Sato polynomial and the jumping coefficients of the multiplier id...

We introduce the notion of Bernstein-Sato polynomial of an arbitrary variety, using the theory of V -filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration of multiplier ideals coincides essentially with the restriction of the V -filtration. This implies a relation between the roots of the Bernstein-Sato polynomial and the jumping coefficients of the multiplier id...

The paper considers the robust stability veriication of polynomials with polynomial parameter dependency. A new algorithm is presented which relies on the expansion of a multivariate polynomial into Bernstein polynomials and is based on the inspection of the value set of the family of polynomials on the imaginary axis. It is shown how an initial interval on the imaginary axis through which zero...

Let k be a field of characteristic 0. Given a polynomial mapping f = (f1, . . . , fp) from kn to kp, the local Bernstein–Sato ideal of f at a point a ∈ kn is defined as an ideal of the ring of polynomials in s = (s1, . . . , sp). We propose an algorithm for computing local Bernstein–Sato ideals by combining Gröbner bases in rings of differential operators with primary decomposition in a polynom...

We use recently calculated next-to-next-to-leading (NNLO) anomalous dimension coefficients for the n = 1, 3, 5, . . . , 13 moments of the xF3 structure function in νN scattering, together with the corresponding three-loop Wilson coefficients, to obtain improved QCD predictions for the moments. The Complete Renormalization Group Improvement (CORGI) approach is used, in which all dependence on re...