Related papers: Computing quadratic function fields with high 3-ra…
We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $\mathbb F_q$, with explicit real multiplication by an order $\mathbb Z[\eta]$ in a totally real cubic…
We study the problem of approximating an unknown function $f:\mathbb{R}\to\mathbb{R}$ by a degree-$d$ polynomial using as few function evaluations as possible, where error is measured with respect to a probability distribution $\mu$.…
Fast algorithms for arithmetic on real or complex polynomials are well-known and have proven to be not only asymptotically efficient but also very practical. Based on Fast Fourier Transform (FFT), they for instance multiply two polynomials…
In probability theory, the partition function is a factor used to reduce any probability function to a density function with total probability of one. Among other statistical models used to represent joint distribution, Markov random fields…
One standard approach to compute the Hilbert function of any graded module over a field is to come up with a free-resolution for the graded module and another is via a Hilbert power series which serves as a generating function. The proposed…
We compute the conformal blocks associated with scalar-scalar-fermion-fermion 4-point functions in 3D CFTs. Together with the known scalar conformal blocks, our result completes the task of determining the so-called `seed blocks' in three…
Accelerating the convergence of second-order optimization, particularly Newton-type methods, remains a pivotal challenge in algorithmic research. In this paper, we extend previous work on the \textbf{Quadratic Gradient (QG)} and rigorously…
Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an…
Using Stickelberger's theorem on Gauss sums, we show that if $F$ is a planar function on a finite field $\mathbb{F}_q$, then for all non-zero functions $G : \mathbb{F}_q \to \mathbb{F}_q$, we have \begin{equation*} d_{\mathsf{alg}}(G \circ…
Let $p$ be an odd prime, $q=p^e$, $e\ge 1$, and $\mathbb{F} = \mathbb{F_q}$ denote the finite field of $q$ elements. Let $f: \mathbb{F}^2\to \mathbb{F}$ and $g: \mathbb{F}^3\to \mathbb{F}$ be functions, and let $P$ and $L$ be two copies of…
Our contribution is a bounded cubic compilation theorem. For each fixed resource parameter $k$, syntactic proof checking at resource level $k$ is faithfully represented by a finite bounded-domain system of cubic polynomial equations. Every…
Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers…
Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the…
An ideal is a classical object of study in the field of algebraic number theory. In maximal quadratic orders of number fields, ideals usually represented by the $\mathbb Z$-basis. This form of representation is used in most of the…
In this paper we prove that there are exactly eight function fields, up to isomorphism, over finite fields with class number one.
This is the first in a series of articles about recovering the full algebraic structure of a boundary conformal field theory (CFT) from the scaling limit of the critical Ising model in slit-strip geometry. Here, we introduce spaces of…
In this thesis, we develop algorithms similar to the Gaussian elimination algorithm in symplectic and split orthogonal similitude groups. As an application to this algorithm, we compute the spinor norm for split orthogonal groups. Also, we…
We present an infinite family of quadratic APN functions on a finite field of dimension over GF(2) divisible by 3.
A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra…
Let $f\in W^{3,1}_{\mathrm{loc}}(\Omega)$ be a function defined on a connected open subset $\Omega\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain…