Recently, Kashaev and the first author constructed an R -matrix from a Nichols algebra with an automorphism, that leads, via the Reshetikhin-Turaev functor, to a multivariable polynomial invariant of knots. Applying this to a rank 2 Nichols algebra, results in a sequence Vn of 2-variable knot polynomials with integer coefficients, the first polynomial been identified with the Links-Gould polynomial. In this note we present the results of the computation of the Vn -polynomials for n=1,2,3,4 . This leads to the discovery of emerging patterns, including the genus bound for V2 being an equality for all 352.2 million knots with at most 19 crossings, as well as unexpected Conway mutations that seem undetected by the Vn -polynomials as well as by Heegaard Floer Homology and Khovanov Homology.
This paper introduces a method for constructing canonical bases of weakly holomorphic modular forms for Gamma 0(N), where N is squarefree and the genus of X0 & lowast;(N) is 0 or 1. Our approach, using Atkin-Lehner involutions, constructs bases for the space Mk!(N), overcoming limitations of previous studies restricted to the space of forms with poles only at specific cusps. We provide explicit constructions of canonical basis {fk,m epsilon} for Mk!,epsilon(N) and investigate arithmetic properties of these bases, including coefficient integrality and duality relations. We also present generating functions for basis elements and extend Ramanujan-style congruences to squarefree levels.
In this paper, we introduce a computational algorithm for decomposing irreducible quasi-ordinary hypersurfaces defined by a minimal polynomial f and a branch zeta with s characteristic exponents. Our approach begins by truncating zeta into a sequence zeta 1,zeta 2,& mldr;,zeta s, where each zeta i retains the first i characteristic exponents and defines a hypersurface Y-i with minimal polynomial fi. We prove that for every i and j satisfying 1 <= i <= s-1 and i+1 <= j <= s, the polynomial f(i) is an i-semi-root of f(j), and we establish support properties of each f(i). The algorithm leverages both the semigroup structure and the support of the involved polynomials through an iterative elimination process. Our results provide a systematic framework for the semi-root decomposition of a truncated quasi-ordinary branch, and we include pseudocode for the algorithm.
An r & times;s matrix is intercalate if all entries in each row are distinct, all entries in each column are distinct, and each 2 & times;2 submatrix has either two or four distinct entries. Yuzvinsky's conjecture on intercalate matrices claims that the number of distinct entries in such an intercalate matrix has a tight lower bound given by the Hopf-Stiefel function r degrees s. This conjecture is known to be true in many cases, including for all r,s <= 16. In this work, we describe a computational approach, including the relevant algorithms and implementations, which allowed us to verify Yuzvinsky's conjecture for all r,s <= 32 and to classify r & times;r symmetric intercalate matrices for all r <= 16.
The famous Littlewood conjecture states that lim inf (q ->infinity)q parallel to q alpha parallel to parallel to q beta parallel to=0 for all alpha,beta is an element of R, where parallel to alpha parallel to denotes the distance of alpha from the nearest integer. Given epsilon>0, we introduce a new algorithm based on the simple continued fraction expansions of alpha and beta, to check whether inf(q is an element of Z+ )q parallel to q alpha parallel to parallel to q beta parallel to
We study the expected number of rolls required for the cumulative sum of a fair six-sided die to first enter a prescribed target set H subset of Z >= 0 . A one-variable dynamic-programming formulation is introduced that removes dependence on the roll count. Within this framework, the infinite process is truncated at a large cutoff N and corrected by an analytically derived overshoot term that accounts for the rare event of exceeding N before entering H . Explicit bounds on this residual yield a strict two-sided estimate of the truncation error. The method is numerically efficient, requiring constant memory and linear time in the cutoff. For the perfect-square target set H={n2:n is an element of N} , all quantities are evaluated explicitly, yielding E[T]=7.07976423755110510389555305690818489468 & mldr;, provably correct to 1017 decimal places. This constitutes the most precise result known to date and establishes a general framework for high-accuracy computation of discrete hitting times.
We establish a higher-dimensional irrationality criterion for periods which are presented as Mellin integrals depending on many parameters. The criterion is stated as an upper bound on the multi-variate transfinite diameter of the image of the domain of integration under the Mellin arguments. Most of the paper is devoted to studying notions of transfinite diameter relative to very general multivariate Vandermonde matrices. As a proof of principle, we illustrate how this approach works with detailed computations in the case of a 5-parameter family of integrals for ζ(2) on ℳ_0,5, the moduli space of curves of genus 0 with 5 marked points. This yields a `higher-dimensional' proof of the irrationality of ζ(2), based on an upper bound for a certain kind of transfinite diameter associated to ℳ_0,5.
Work of Bourdon-Clark-Pollack shows that the set of degrees d is an element of Z+ for which the classification of torsion subgroups of CM elliptic curves over all number fields of degree d is the same as the classification of CM torsion of elliptic curves over Q has positive asymptotic density. Based on this, I conjectured that for every d0 is an element of Z+, the set of d is an element of Z+ such that the classification of CM torsion in degree d as the same as in degree d0 has positive density. This was proven for all odd d0 by Bourdon-Pollack in 2017. Here we give the first results on even d0: the conjecture holds for d0=2 and for d0=2p0 for a set of primes p0 of relative density one. However, we will also explain why the conjecture seems likely to be false for d0=2p0 where p0 lies in an infinite set of prime numbers, including p0=3.
The aim of this paper is to prove that the Word Problem and the Conjugacy Problem for the structure left skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation are solvable. In order to achieve this result, we need to introduce the concept of (almost) polycyclic left skew brace and to develop a general theory showing that almost polycyclic left skew braces are controlled by their finite homomorphic images.Our results provide us with the first class of infinite solutions of the Yang-Baxter equation on which is possible to work in an algorithmic waythe class of almost polycyclic solutions.
In this paper, we investigate idempotent hom-groups and their classification. We demonstrate that every idempotent hom-group gives rise to a shelf structure, which in turn yields set-theoretic solutions to the Yang-Baxter equation. We classify idempotent hom-groups of low order, a result achieved through the extension theory developed in this work, with the aid of computational algebra tools.
Recently, Baake and Coons proved several results on the average size of the autocorrelations of the Thue-Morse sequence. They also considered the absolute value of the autocorrelations, and showed that the average value of the autocorrelations is zero. In particular, they showed that & sum;(n <= x)|eta(n)|=o(x(alpha)) for any alpha> log (3)/ log (4). In this paper, we sharpen this result, providing upper and lower bounds for alpha. On the way to our lower bounds, we obtain the structure of the linear representation of the point-wise product of two k-regular sequences, which may be of independent interest.
Let A be a 2 & times;2 matrix over a finite field and consider the Yang-Baxter matrix equation XAX=AXA with respect to A. We use a method of computational ideal theory to explore the geometric structure of the affine variety of all solutions to this equation. In particular, we exhibit all solutions explicitly and determine cardinality formulas for these varieties.
In this paper, we introduce a new depicting of the so-called numerical semigroup tree T. By exploring computationally this improved picture, relying on the type notion of a semigroup, we found that the number of semigroups of genus g and type t is constant when t is close to g while g grows. We also study the unimodality of various sequences as well as the behavior of the leaves in T. We put forward several conjectures that are supported by various computational experiments.
Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences (an)n >= 0 of primitive eta-products that satisfy the generic convolutive property & sum;n >= 0amnqn=(& sum;n >= 0anqn)m for a specific positive integer m. Given the results of an exhaustive search of the Online Encyclopedia of Integer Sequences for such sequences for m up to 6, we first focus on the case where m=2 with our attention mainly paid to the combinatorics of two 2-convolutive sequences, featuring bijective proofs for both. For other 2-convolutive sequences discovered in the OEIS, we apply generating function manipulations to show their convolutivity. We also give two examples of 3-convolutive sequences. Finally, we discuss other convolutive series that are not eta-products.
We investigate Eisenstein discriminants, which are squarefree integers d equivalent to 5(mod8) such that the fundamental unit epsilon d of the real quadratic field K=Q(d) satisfies epsilon d equivalent to 1(mod2OK). These discriminants are related to a classical question of Eisenstein and have connections to the class groups of orders in quadratic fields as well as to real cubic fields. We present numerical computations of Eisenstein discriminants up to 1011, suggesting that their counting function up to x is approximated by NE(x)approximate to 13 pi 2x-0.024x5/6. This supports a conjecture of Stevenhagen while revealing a surprising secondary term, which is similar to (but subtly different from) the secondary term in the counting function of real cubic fields. We include technical details of our computation method, which uses a modified infrastructure approach implemented on GPUs.
The homology of free Lie algebras with coefficients in tensor products of the adjoint representation working over Q contains important information on the homological properties of polynomial outer functors on free groups. The latter category was introduced in joint work with Vespa, motivated by the study of higher Hochschild homology of wedges of circles. There is a splitting of this homology by polynomial degree (for polynomiality with respect to the generators of the free Lie algebra) and one can consider the polynomial degree relative to the number of tensor factors in the coefficients. It suffices to consider the Lie algebra homology in homological degree one; this vanishes in relative degree 0 and is readily calculated in relative degree 1. This paper calculates the homology in relative degree 2, which presents interesting features. This confirms a conjecture of Gadish and Hainaut.
We give a nonnegative step function with 575 equally spaced intervals such that & Vert;f & lowast;f & Vert;L2(R)2 & Vert;f & lowast;f & Vert;L infinity(R)& Vert;f & lowast;f & Vert;L1(R)>= 0.901564. This improves upon a recent result of Deepmind's AlphaEvolve from May 2025, which found a nonnegative step function with 50 equally space intervals for which the left hand side is >= 0.8962. Our function was found using simulated annealing and gradient based methods.
We show that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners. We also determine that there are exactly 3 primitive tight instances of the conjecture, only two of which are tight for the non-shifted conjecture (LRC). Our proof is computational, relying on a rephrasing of the sLRC in terms of covering radii of certain zonotopes (Henze and Malikiosis, 2017), and on an upper bound for the (integer) velocities to be checked (Malikiosis, Santos and Schymura, 2024+). As a tool for the proof, we devise an algorithm for bounding the covering radius of rational lattice polytopes, based on constructing dyadic fundamental domains.
Let H-g denote the coarse moduli space of smooth hyperelliptic curves of genus g in characteristic p >= 3, and let H(g)f denote the p-rank f stratum of H-g for 0 <= f <= g. Achter and Pries note that determining the number of irreducible components of H(g)f would lead to several intriguing corollaries. In this paper, we present a computational approach for estimating the number of irreducible components in various p-rank strata. Our strategy involves sampling curves over finite fields and calculating their p-ranks. From the data gathered, we conjecture that the non-ordinary locus is geometrically irreducible for all genera g>1. The data also leads us to conjecture that the moduli space H-g(g-2) is irreducible and suggests that H-g(f) is irreducible for all 1 <= f <= g. We conclude with a brief discussion on H-g(0).
Denote f(n) := Sigma(1 <= k <= n) tau(2(k) - 1), where tau is the number of divisors function. Motivated by a question of Paul Erdos, we show that the sequence of ratios f(2n)/f(n) is unbounded. We also present conditional results on the divergence of this sequence to infinity. Finally, we give experimental evidence on both the conjecture f(2n)/f(n) ->infinity and our sufficient conditions for it to hold
