Related papers: Efficient algorithms for computing the Euler-Poinc…
We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In…
Computation of (approximate) polynomials common factors is an important problem in several fields of science, like control theory and signal processing. While the problem has been widely studied for scalar polynomials, the scientific…
We present a general technique, based on parametric search with some twist, for solving a variety of optimization problems on a set of semi-algebraic geometric objects of constant complexity. The common feature of these problems is that…
We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…
The discrete logarithm problem (DLP) generalizes to the constrained DLP, where the secret exponent $x$ belongs to a set known to the attacker. The complexity of generic algorithms for solving the constrained DLP depends on the choice of the…
We give several families of polynomials which are related by Eulerian summation operators. They satisfy interesting combinatorial properties like being integer-valued at integral points. This involves nearby-symmetries and a recursion for…
Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a $d^{p+q}$-dimensional matrix variable that…
Let $V$ be the set of real common solutions to $F = (f_1, \ldots, f_s)$ in $\mathbb{R}[x_1, \ldots, x_n]$ and $D$ be the maximum total degree of the $f_i$'s. We design an algorithm which on input $F$ computes the dimension of $V$. Letting…
This paper considers the minimization problem of relaxed submodular functions. For a positive integer $k$, a set function is called $k$-distant submodular if the submodular inequality holds for every pair whose symmetric difference is at…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
The EM (Expectation-Maximization) algorithm is regarded as an MM (Majorization-Minimization) algorithm for maximum likelihood estimation of statistical models. Expanding this view, this paper demonstrates that by choosing an appropriate…
In this paper we study the problem of computing a Kolchin characteristic set of a radical differential ideal. The central part of the article is the presentation of algorithms solving this problem in two principal cases: for ordinary…
Polynomial ensembles are determinantal point processes associated with (non necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
We address complexity issues for linear differential equations in characteristic $p>0$: resolution and computation of the $p$-curvature. For these tasks, our main focus is on algorithms whose complexity behaves well with respect to $p$. We…
Barvinok introduced the symmetrized determinant ($\sdet$) as a \emph{non-commutative} analogue of the determinant. Intuitively, given a square matrix over an associative algebra, we can obtain the symmetrized determinant by averaging over…
Consider a system of n polynomial equations and r polynomial inequations in n indeterminates of degree bounded by d with coefficients in a polynomial ring of s parameters with rational coefficients of bit-size at most $\sigma$. From the…
We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of…
The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by [Thomassen '89 & '93], and it is known to be fixed-parameter tractable. However, the…
For any $\ell > 0$, we present an algorithm which takes as input a semi-algebraic set, $S$, defined by $P_1 \leq 0,...,P_s \leq 0$, where each $P_i \in \R[X_1,...,X_k]$ has degree $\leq 2,$ and computes the top $\ell$ Betti numbers of $S$,…