Related papers: On computing the Hermite form of a matrix of diffe…
In this article, we give evidence that computing Fourier coefficients of the Hecke eigenforms for composite indices is no easier than factoring integers. In particular, we show that the existence of a polynomial time algorithm that, given…
In this work, based on quantum operator Hermite polynomials and Weyl's mapping rule, we find a generation function of the two-variable Hermite polynomials. And then, noting that the Weyl ordering is invariant under the similar…
We give two widest Mehler's formulas for the univariate complex Hermite polynomials $H_{m,n}^\nu$, by performing double summations involving the products $u^m H_{m,n}^\nu (z,\overline{z}) \overline{H_{m,n}^\nu (w,\overline{w})}$ and $u^m…
The Hermite Normal Form (HNF) is a canonical representation of matrices over any principal ideal domain. Over the integers, the distribution of the HNFs of randomly looking matrices is far from uniform. The aim of this article is to present…
Let $\mathcal{R} = \mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$ of characteristic 0. Consider $n$ algebraically independent elements $g_1, \dots, g_n$ in $\mathcal{R}$. Let $\mathcal{S}$ denote…
Testing the unimodular equivalence of two full-dimensional integral simplices can be reduced to testing unimodular permutation (UP) equivalence of two nonsingular matrices. We conduct a systematic study of UP-equivalence, which leads to the…
In this article, the matrix elements of a representation of the 5-dimensional Lie algebra K5 are obtained for the first time. The bivariate degenerate Hermite polynomials Hm(z1, z2|{\tau} ) are considered within the context of this…
We construct explicit differential operators on hermitian modular forms, extending methods developed for Siegel modular forms. These differential operators are closely related to the two-variable spherical pluriharmonic polynomials. We…
We present an accurate investigation of the algebraic conditions that the symbols of a univariate, binary, Hermite subdivision scheme have to fulfil in order to reproduce polynomials. These conditions are sufficient for the scheme to…
The Hermite interpolation formulas are based on the interpretation of interpolation nodes as roots of suitable polynomials. Therefore, such formulas belong to the class of algebraic interpolations. The article considers a multidimensional…
Inspired by the work about solutions of a system of real polynomial equations done by Hermite, this paper introduces a Hermitian form, which encodes information about solutions of a system of complex polynomial equations with conjugate…
We collect here elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange \& Hermite…
Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…
The multiplication theorem for univariate Hermite polynomials $H_k(\lambda x)$ is well-known. In this paper we generalize this result to multivariate Hermite polynomials ${\rm H}_{\bf k}({\mathbf{\Lambda}}{\bf x};{\mathbf{\Sigma}})$, and…
This note presents the multivariate Hermite criterion: a practical and powerful algorithm for determining the number of distinct real and complex roots of a zero-dimensional system of polynomials in any finite number of variables. The final…
We derive iterative methods for computing the Fr\'{e}chet derivative of the map which sends a full-rank matrix $A$ to the factor $U$ in its polar decomposition $A=UH$, where $U$ has orthonormal columns and $H$ is Hermitian positive…
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
In this paper, we develop general machinery for computing the classifying ring $L^A$ of one-dimensional formal $A$-modules, for various commutative rings $A$. We then apply the machinery to obtain calculations of $L^A$ for various number…
We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant…
We design a new algorithm for solving parametric systems having finitely many complex solutions for generic values of the parameters. More precisely, let $f = (f_1, \ldots, f_m)\subset \mathbb{Q}[y][x]$ with $y = (y_1, \ldots, y_t)$ and $x…