For an isolated hypersurface singularity f:(Cn,0)->(C,0) , we study the & Lstrok;ojasiewicz exponent & lstrok;(f) of f . We formulate three conjectures concerning: (i) attainment of & lstrok;(f) on coordinate polar curves, (ii) topological invariance of & lstrok;(f) , and (iii) a local Newton-diagram formula for & lstrok;(f) in the Kushnirenko non-degenerate case. We compute & lstrok;(f) for all singularities of modality at most 2 in Arnold's list and for several further families, obtaining systematic evidence in favor of these conjectures. We also find to what extent the collection of numerical invariants {modality(f),& lstrok;(f),mu(f),corank(f),stabord(f)} determines the stable equivalence class of f for the aforementioned singularities.
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.
