Related papers: Fast Computation of Generalized Dedekind Sums
Fourier-Dedekind sums are a generalization of Dedekind sums - important number-theoretical objects that arise in many areas of mathematics, including lattice point enumeration, signature defects of manifolds and pseudo random number…
We present a new GCD algorithm of two integers or polynomials. The algorithm is iterative and its time complexity is still $O(n \\log^2 n ~ log \\log n)$ for $n$-bit inputs.
A method is described which allows to evaluate efficiently a polynomial in a (possibly trivial) extension of the finite field of its coefficients. Its complexity is shown to be lower than that of standard techniques when the degree of the…
Subset Sum is a classical optimization problem taught to undergraduates as an example of an NP-hard problem, which is amenable to dynamic programming, yielding polynomial running time if the input numbers are relatively small. Formally,…
The paper is devoted to construction of some closed inductive sequence of models of the generalized second-order Dedekind theory of real numbers with exponentially increasing powers. These models are not isomorphic whereas all models of the…
Numerical data processing is a key task across different fields of computer technology use. However, even simple summation of values is not precise due to the floating point representation use. This paper presents a practical algorithm for…
Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in a fine-grained manner, we deviate from…
We show how to use extended word series in the reduction of continuous and discrete dynamical systems to normal form and in the computation of formal invariants of motion in Hamiltonian systems. The manipulations required involve complex…
This paper introduces a general technique for estimating the absolute value of pure Gaussian sums of order k over a prime p for a class of composite order k. The new estimate improves the classical estimate by a factor of about 2 or better…
Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…
We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of…
We give several new algorithms for dense polynomial multiplication based on the Kronecker substitution method. For moderately sized input polynomials, the new algorithms improve on the performance of the standard Kronecker substitution by a…
Dedekind sums, arithmetic correlation sums that arose in Dedekind's study of the modular transformation of the logarithm of the eta-function, are surprisingly ubiquitous. Their arithmetic properties attracted the attention of number…
We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and…
We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomized algorithm for the following problem: If a homogeneous polynomial $f \in K[x_1 , . . . , x_n]$ (where $K \subseteq \mathbb{C}$) of…
Gradient-based algorithms, popular strategies to optimization problems, are essential for many modern machine-learning techniques. Theoretically, extreme points of certain cost functions can be found iteratively along the directions of the…
This paper introduces a novel general-purpose algorithm for Pauli decomposition that employs matrix slicing and addition rather than expensive matrix multiplication, significantly accelerating the decomposition of multi-qubit matrices. In a…
We study the decomposition of multivariate polynomials as sums of powers of linear forms. As one of our main results we give an algorithm for the following problem: given a homogeneous polynomial of degree 3, decide whether it can be…
Given a multiset $S$ of $n$ positive integers and a target integer $t$, the Subset Sum problem asks to determine whether there exists a subset of $S$ that sums up to $t$. The current best deterministic algorithm, by Koiliaris and Xu…
This paper develops new combinatorial approaches to analyze and compute special set partitions, called complementary set partitions, which are fundamental in the study of generalized cumulants. Moving away from traditional graph-based and…