Related papers: Efficient Quotients of Non-Commutative Polynomials
We present an algorithmic equivalent statement to the Jacobian conjecture. Given a polynomial map F on an affine space of dimension n, our algorithm constructs n sequences of polynomials such that F is invertible if and only if the zero…
We consider the quantum complexity of estimating matrix elements of unitary irreducible representations of groups. For several finite groups including the symmetric group, quantum Fourier transforms yield efficient solutions to this…
If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional…
We give an effective method to compute the entropy for polynomials orthogonal on a segment of the real axis that uses as input data only the coefficients of the recurrence relation satisfied by these polynomials. This algorithm is based on…
We propose a computationally efficient method to solve the dynamics of operators of bosonic quantum systems coupled to their environments. The method maps the operator under interest to a set of complex-valued functions, and its adjoint…
This paper deals with circulant matrices. It is shown that a circulant matrix can be multiplied by a vector in time O(n log(n)) in a ring with roots of unity without making use of an FFT algorithm. With our algorithm we achieve a speedup of…
We study the problem of representing multivariate polynomials with rational coefficients, which are nonnegative and strictly positive on finite semialgebraic sets, using rational sums of squares. We focus on the case of finite semialgebraic…
We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further…
In this paper we study the complexity of factorization of polynomials in the free noncommutative ring $\mathbb{F}\langle x_1,x_2,\dots,x_n\rangle$ of polynomials over the field $\mathbb{F}$ and noncommuting variables $x_1,x_2,\ldots,x_n$.…
The study of images of noncommutative polynomials on algebras has attracted considerable attention. We investigate polynomial images and the additive structures they generate in associative algebras, focusing on sums and products of values.…
In many high-dimensional problems,polynomial-time algorithms fall short of achieving the statistical limits attainable without computational constraints. A powerful approach to probe the limits of polynomial-time algorithms is to study the…
We describe an algorithm for using a quantum computer to calculate mean values of observables and the partition function of a quantum system. Our algorithm includes two sub-algorithms. The first sub-algorithm is for calculating, with…
There exists a wide variety of efficiency methods for natural language processing (NLP) tasks, such as pruning, distillation, dynamic inference, quantization, etc. We can consider an efficiency method as an operator applied on a model.…
We provide a general algorithm for the computation of the unramified Brauer group of quotients of rational varieties by finite groups.
This paper introduces a non-variational quantum algorithm designed to solve a wide range of combinatorial optimisation problems, including constrained and non-binary problems. The algorithm leverages an engineered interference process…
Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…
Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. Combining existing methods with recent work…
We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…