Related papers: Classical and Quantum Polynomial Reconstruction vi…

In this paper we describe an algorithm for the computation of canonical forms of finite subsets of $\mathbb{Z}^d$, up to affinities over $\mathbb{Z}$. For fixed dimension $d$, this algorithm has worst-case asymptotic complexity $O(n \log^2…

We apply a symbolic approach of the general quadratic decomposition of polynomial sequences - presented in a previous article referenced herein - to polynomial sequences fulfilling specific orthogonal conditions towards two given…

Solving the electronic structure problem of molecules and solids to high accuracy is a major challenge in quantum chemistry and condensed matter physics. The rapid emergence and development of quantum computers offer a promising route to…

We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…

Let $f, f_1, \ldots, f_\nV$ be polynomials with rational coefficients in the indeterminates $\bfX=X_1, \ldots, X_n$ of maximum degree $D$ and $V$ be the set of common complex solutions of $\F=(f_1,\ldots, f_\nV)$. We give an algorithm…

We study the problem of finding elements in the intersection of an arbitrary conic variety in $\mathbb{F}^n$ with a given linear subspace (where $\mathbb{F}$ can be the real or complex field). This problem captures a rich family of…

We present an algorithm which computes the $D^{th}$ term of a sequence satisfying a linear recurrence relation of order $d$ over a field $K$ in $O( \mathsf{M}(\bar d)\log(D) + \mathsf{M}(d)\log(d))$ operations in $K$, where $\bar d \leq d$…

Let $k$ be a locally compact complete field with respect to a discrete valuation $v$. Let $\oo$ be the valuation ring, $\m$ the maximal ideal and $F(x)\in\oo[x]$ a monic separable polynomial of degree $n$. Let $\delta=v(\dsc(F))$. The…

Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly…

We continue to investigate binary sequence $(f_u)$ over $\{0,1\}$ defined by $(-1)^{f_u}=\left(\frac{(u^w-u^{wp})/p}{p}\right)$ for integers $u\ge 0$, where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol and we restrict…

Let C be a depth-3 circuit with n variables, degree d and top fanin k (called sps(k,d,n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically…

Difference sets are basic combinatorial structures that have applications in signal processing, coding theory, and cryptography. We consider the problem of identifying a shifted version of the characteristic function of a (known) difference…

In this letter, we consider a problem of reconstructing an unknown discrete signal taking values in a finite alphabet from incomplete linear measurements. The difficulty of this problem is that the computational complexity of the…

In this paper, monic polynomials orthogonal with deformation of the Freud-type weight function are considered. These polynomials fullfill linear differential equation with some polynomial coefficients in their holonomic form. The aim of…

Vector set orthogonal normalization and matrix QR decomposition are fundamental problems in matrix analysis with important applications in many fields. We know that Gram-Schmidt process is a widely used method to solve these two problems.…

We present a new algorithm for the computation of the irreducible factors of degree at most $d$, with multiplicity, of multivariate lacunary polynomials over fields of characteristic zero. The algorithm reduces this computation to the…

We present a quantum algorithm for solving the hidden subgroup problem in the general linear group over a finite field where the hidden subgroup is promised to be a conjugate of the group of the invertible lower triangular matrices. The…

Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations…

We prove that quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating the value of simple sums, quadratically signed weight enumerators. The problem of estimating these sums can be…

We consider the task of verifying the correctness of quantum computation for a restricted class of circuits which contain at most two basis changes. This contains circuits giving rise to the second level of the Fourier Hierarchy, the lowest…