Related papers: Computing the Newton polygon of the implicit equat…
We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over…
A method is presented for computing all the affine equivalences between two rational ruled surfaces defined by rational parametrizations that works directly in parametric rational form, i.e. without computing or making use of the implicit…
For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane…
This paper proposed a method to judge whether the point is inside or outside of the simple convex polygon by the intersection of the vertical line. It determined the point to an area enclosed by two straight lines, then convert the problem…
Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block and G\"ottsche, and further extended by…
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows…
We present algorithms for classifying rational polygons with fixed denominator and number of interior lattice points. Our approach is to first describe maximal polygons and then compute all subpolygons, where we eliminate redundancy by a…
For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input…
Neural networks have recently been used to analyze diverse physical systems and to identify the underlying dynamics. While existing methods achieve impressive results, they are limited by their strong demand for training data and their weak…
We propose a novel numerical approach to compute the Pareto front in multivariate polynomial multi-objective optimization problems. When the objective functions and (equality) constraints are multivariate polynomials, the Pareto front,…
We introduce an efficient combination of polyhedral analysis and predicate partitioning. Template polyhedral analysis abstracts numerical variables inside a program by one polyhedron per control location, with a priori fixed directions for…
Recent efforts to unravel the mystery of implicit regularization in deep learning have led to a theoretical focus on matrix factorization -- matrix completion via linear neural network. As a step further towards practical deep learning, we…
The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…
We determine all modular curves $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from $X_0(N)$ to a positive rank elliptic…
In this paper, we present a Newton-like method based on model reduction techniques, which can be used in implicit numerical methods for approximating the solution to ordinary differential equations. In each iteration, the Newton-like method…
Recent advances in the integration of deep learning with automated theorem proving have centered around the representation of logical formulae as inputs to deep learning systems. In particular, there has been a growing interest in adapting…
Even when neural networks are widely used in a large number of applications, they are still considered as black boxes and present some difficulties for dimensioning or evaluating their prediction error. This has led to an increasing…
The ruled surface is a typical modeling surface in computer aided geometric design. It is usually given in the standard parametric form. However, it can also be in the forms than the standard one. For these forms, it is necessary to…
We present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations. The algorithms are based on the existence of a linear relationship between two…
In this paper I demonstrate that any pair (m, n) of non-zero and distinct rational numbers may have, at most, four representations as the product of two rational factors such that the sum of factors of m coincides with the sum of factors of…