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Braid monodromy is an important tool for computing invariants of curves and surfaces. In this paper, the \emph{rectangular braid diagram (RBD)} method is proposed to compute the braid monodromy of a completely reducible $n$-gonal curve,…
We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem.…
We give a complete characterization of degree two rational maps with potential good reduction over local fields. We show this happens exactly when the map corresponds to an integral point in the moduli space. We detail an algorithm by which…
Many combinatorial optimisation problems hide algebraic structures that, once exposed, shrink the search space and improve the chance of finding the global optimal solution. We present a general framework that (i) identifies algebraic…
In two previous papers we showed that any analytically integrable vector field admits a local analytic Poincar\'e-Birkhoff normalization in the neighborhood of a singular point. The aim of this paper is to extend this analytic normalization…
We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three…
Bilevel optimization (BO) is useful for solving a variety of important machine learning problems including but not limited to hyperparameter optimization, meta-learning, continual learning, and reinforcement learning. Conventional BO…
Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…
This paper investigates the p-adic valuation trees of degree-2 and degree-3 polynomials in two variables over any prime p, building upon prior research outlined in [14].
The bivariate difference filed $(\mathbb{F}(\alpha, \beta), \sigma)$ provides an algebraic framework for a sequence satisfying a recurrence of order two and it could transform the summation involving a sequence satisfying a recurrence of…
We prove an inversion formula for summatory arithmetic functions. As an application, we obtain an arithmetic relationship between summatory Piltz divisor functions and a sum of the M\"obius function over certain integers, denoted by…
We study the complexity of solving the \emph{generalized MinRank problem}, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most $r$. A natural algebraic representation of this problem gives rise to a…
Normal multi-scale transform [4] is a nonlinear multi-scale transform for representing geometric objects that has been recently investigated [1, 7, 10]. The restrictive role of the exact order of polynomial reproduction $P_e$ of the…
We describe a framework for reformulating and solving optimization problems that generalizes the well-known framework originally introduced by Benders. We discuss details of the application of the procedures to several classes of…
Multi-domain generalization (mDG) is universally aimed to minimize the discrepancy between training and testing distributions to enhance marginal-to-label distribution mapping. However, existing mDG literature lacks a general learning…
Let $V = < x^{\lambda_i}p_{ij}(x), i=1,...,n, j=1, ..., N_i > $ be a space of quasi-polynomials in $x$ of dimension $N=N_1+...+N_n$. The regularized fundamental differential operator of $V$ is the polynomial differential operator…
A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $\Omega$ is a general smooth domain with a curved…
We consider the problem of translating between irreducible closed sets and implicational bases in closure systems. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph…
We propose a general reduction procedure for classical field theories provided with abelian gauge symmetries in a Lagrangian setting. These ideas come from an axiomatic presentation of the general boundary formulation (GBF) of field…
In previous literature, backward error analysis was used to find ordinary differential equations (ODEs) approximating the gradient descent trajectory. It was found that finite step sizes implicitly regularize solutions because terms…