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We give algorithms for computing multiplier ideals using Gr\"obner bases in Weyl algebras. The algorithms are based on a newly introduced notion which is a variant of Budur--Musta\c{t}\v{a}--Saito's (generalized) Bernstein--Sato polynomial.…
In the long paper "Family Blowup formula, Admissible Graphs and the Enumeration of Singular Curves (I)" (appearing in JDG), the author solved the enumeration problem of nodal (or general singular) curve counting on algebraic surfaces by…
We study the $k$-median clustering problem for high-dimensional polygonal curves with finite but unbounded number of vertices. We tackle the computational issue that arises from the high number of dimensions by defining a…
In this paper, we explore some properties of Galois hulls of cyclic serial codes over a chain ring and we devise an algorithm for computing all the possible parameters of the Euclidean hulls of that codes. We also establish the average…
The "defect" of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre bound for the curve. We present a construction for producing genus-4 double covers of genus-2 curves over…
The generalized winding number is an essential part of the geometry processing toolkit, allowing to quantify how much a given point is inside a surface, even when the surface has boundaries and noise. We propose a new universal method to…
We consider the problem of subspace clustering: given points that lie on or near the union of many low-dimensional linear subspaces, recover the subspaces. To this end, one first identifies sets of points close to the same subspace and uses…
We design a probabilistic algorithm for computing endomorphism rings of ordinary elliptic curves defined over finite fields that we prove has a subexponential runtime in the size of the base field, assuming solely the generalized Riemann…
A simple and efficient algorithm to numerically compute the genus of surfaces of three-dimensional objects using the Euler characteristic formula is presented. The algorithm applies to objects obtained by thresholding a scalar field in a…
We propose a novel way of computing surface folding maps via solving a linear PDE. This framework is a generalization to the existing quasiconformal methods and allows manipulation of the geometry of folding. Moreover, the crucial quantity…
We review the Shimura-Taniyama method for computing the Newton polygon of an abelian variety with complex multiplication. We apply this method to cyclic covers of the projective line branched at three points. As an application, we produce…
We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (sub-groupoid), and only counts…
We prove the quasimodularity of generating functions for counting pillowcase covers, with and without Siegel-Veech weight. Similar to prior work on torus covers, the proof is based on analyzing decompositions of half-translation surfaces…
We present an algorithm which speeds scalar multiplication on a general elliptic curve by an estimated 3.8 % to 8.5 % over the best known general methods when using affine coordinates. This is achieved by eliminating a field multiplication…
We study the $K$-theoretic enumerative geometry of cyclic Nakajima quiver varieties, with particular focus on $\text{Hilb}^{m}([\mathbb{C}^{2}/\mathbb{Z}_{l}])$, the equivariant Hilbert scheme of points on $\mathbb{C}^2$. The direct sum…
The algorithm of Edelsbrunner for surface reconstruction by ``wrapping'' a set of points in R^3 is described.
In this paper we obtain an explicit formula for the number of degree d curves in two dimensional complex projective space, passing through (d(d+3)/2 -k) generic points and having a codimension k singularity, where k is at most 7. In the…
We show that Boolean matrix multiplication, computed as a sum of products of column vectors with row vectors, is essentially the same as Warshall's algorithm for computing the transitive closure matrix of a graph from its adjacency matrix.…
The theory of coverings of the two-dimensional torus is a standard part of algebraic topology and has applications in several topics in string theory, for example, in topological strings. This paper initiates applications of this theory to…
We present an efficient algorithm to compute the Euler factor of a genus 2 curve C/Q at an odd prime p that is of bad reduction for C but of good reduction for the Jacobian of C (a prime of ``almost good'' reduction). Our approach is based…