arXiv subjects

Laurent Busé

Publications and source records attributed to Laurent Busé.

At least 19 recordsLinked to original sources

arXiv2026

Efficient closed-form approaches for pose estimation using Sylvester forms

Solving non-linear least-squares problem for pose estimation (rotation and translation) is often a time consuming yet fundamental problem in several real-time computer vision applications. With an adequate rotation parametrization, the optimization problem can be reduced to the solution of a~system of polynomial equations and solved in closed form. Recent advances in efficient closed form solvers utilizing resultant matrices have shown a promising research direction to decrease the computation time while preserving the estimation accuracy. In this paper, we propose a new class of resultant-based solvers that exploit Sylvester forms to further reduce the complexity of the resolution. We demonstrate that our proposed methods are numerically as accurate as the state-of-the-art solvers, and outperform them in terms of computational time. We show that this approach can be applied for pose estimation in two different types of problems: estimating a pose from 3D to 3D correspondences, and estimating a pose from 3D points to 2D points correspondences.

cs.CV↗

arXiv2025

Generalized weight properties of resultants and discriminants, and applications to projective enumerative geometry

The goal of this text is to understand and prove a formula stated by Salmon, which gives the first terms of some Taylor expansion of the discriminant of a plane algebraic curve. Salmon uses his formula to derive various enumerative quantities for surfaces in $\mathbf{P}^3$. We provide complete proofs of this formula and its enumerative applications, and extend Salmon's considerations to hypersurfaces in a projective space of arbitrary dimension. To this end, we introduce the concept of reduced discriminant, and provide a thorough study of its weight properties; the latter are deeply linked to projective enumerative geometric properties.

math.AG↗

arXiv2025

Solving bihomogeneous polynomial systems with a zero-dimensional projection

We study bihomogeneous systems defining, non-zero dimensional, biprojective varieties for which the projection onto the first group of variables results in a finite set of points. To compute (with) the 0-dimensional projection and the corresponding quotient ring, we introduce linear maps that greatly extend the classical multiplication maps for zero-dimensional systems, but are not those associated to the elimination ideal; we also call them multiplication maps. We construct them using linear algebra on the restriction of the ideal to a carefully chosen bidegree or, if available, from an arbitrary Gröbner bases. The multiplication maps allow us to compute the elimination ideal of the projection, by generalizing FGLM algorithm to bihomogenous, non-zero dimensional, varieties. We also study their properties, like their minimal polynomials and the multiplicities of their eigenvalues, and show that we can use the eigenvalues to compute numerical approximations of the zero-dimensional projection. Finally, we establish a single exponential complexity bound for computing multiplication maps and Gröbner bases, that we express in terms of the bidegrees of the generators of the corresponding bihomogeneous ideal.

math.AC↗

arXiv2025

Bigraded Castelnuovo-Mumford regularity and Gröbner bases

We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal $I$ in the coordinate ring of the product of two projective spaces and the bidegrees of a Gröbner basis of $I$ with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single-graded case, Bayer and Stillman unraveled all aspects of this relationship forty years ago and these results led to complexity estimates for computations with Gröbner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Gröbner bases for $I$. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of $I$.

math.AC↗

arXiv2025

Construction of birational trilinear volumes via tensor rank criteria

We provide effective methods to construct and manipulate trilinear birational maps $ϕ:(\mathbb{P}^1)^3\dashrightarrow \mathbb{P}^3$ by establishing a novel connection between birationality and tensor rank. These yield four families of nonlinear birational transformations between 3D spaces that can be operated with enough flexibility for applications in computer-aided geometric design. More precisely, we describe the geometric constraints on the defining control points of the map that are necessary for birationality, and present constructions for such configurations. For adequately constrained control points, we prove that birationality is achieved if and only if a certain $2\times 2\times 2$ tensor has rank one. As a corollary, we prove that the locus of weights that ensure birationality is $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$. Additionally, we provide formulas for the inverse $ϕ^{-1}$ as well as the explicit defining equations of the irreducible components of the base loci. Finally, we introduce a notion of "distance to birationality" for trilinear rational maps, and explain how to continuously deform birational maps.

math.AG↗

arXiv2024

Toric Sylvester forms

In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety $X$ with respect to the irrelevant ideal of $X$. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on $X$. In particular, we prove that toric Sylvester forms yield bases of some graded components of $I^{\text{sat}}/I$, where $I$ denotes an ideal generated by $n+1$ generic forms, $n$ is the dimension of $X$ and $I^{\text{sat}}$ the saturation of $I$ with respect to the irrelevant ideal of the Cox ring of $X$. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory over smooth toric varieties: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we explote a new formula for computing toric residues of the product of two forms.

math.AG↗

arXiv2022

Tri-linear birational maps in dimension three

A tri-linear rational map in dimension three is a rational map $ϕ: (\mathbb{P}_\mathbb{C}^1)^3 \dashrightarrow \mathbb{P}_\mathbb{C}^3$ defined by four tri-linear polynomials without a common factor. If $ϕ$ admits an inverse rational map $ϕ^{-1}$, it is a tri-linear birational map. In this paper, we address computational and geometric aspects about these transformations. We give a characterization of birationality based on the first syzygies of the entries. More generally, we describe all the possible minimal graded free resolutions of the ideal generated by these entries. With respect to geometry, we show that the set $\mathfrak{Bir}_{(1,1,1)}$ of tri-linear birational maps, up to composition with an automorphism of $\mathbb{P}_\mathbb{C}^3$, is a locally closed algebraic subset of the Grassmannian of $4$-dimensional subspaces in the vector space of tri-linear polynomials, and has eight irreducible components. Additionally, the group action on $\mathfrak{Bir}_{(1,1,1)}$ given by composition with automorphisms of $(\mathbb{P}_\mathbb{C}^1)^3$ defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations.

math.AG↗

arXiv2022

Multigraded Sylvester forms, Duality and Elimination Matrices

In this paper we study the equations of the elimination ideal associated with $n+1$ generic multihomogeneous polynomials defined over a product of projective spaces of dimension $n$. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.

math.AC↗

arXiv2022

Formulas for the eigendiscriminants of ternary and quaternary forms

A $d$-dimensional tensor $A$ of format $n\times n\times \cdots \times n$ defines naturally a rational map $Ψ$ from the projective space $\mathbb{P}^{n-1}$ to itself and its eigenscheme is then the subscheme of $\mathbb{P}^{n-1}$ of fixed points of $Ψ$. The eigendiscriminant is an irreducible polynomial in the coefficients of $A$ that vanishes for a given tensor if and only if its eigenscheme is singular. In this paper we contribute two formulas for the computation of eigendiscriminants in the cases $n=3$ and $n=4$. In particular, by restriction to symmetric tensors, we obtain closed formulas for the eigendiscriminants of plane curves and surfaces in $\mathbb{P}^3$ as the ratio of some determinants of resultant matrices.

math.AG↗

arXiv2021

The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in $\mathbb{P}^n$

For a reduced hypersurface $V(f) \subseteq \mathbb{P}^n$ of degree $d$, the Castelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understood when $V(f)$ is smooth, as well as when $V(f)$ has isolated singularities. We study the regularity of $M(f)$ when $V(f)$ has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by $(d-2)(n+1)$, which is the degree of the Hessian polynomial of $f$. However, this is not always the case, and we prove that in $\mathbb{P}^n$ the regularity of the Milnor algebra can grow quadratically in $d$.

math.AG↗

arXiv2021

Fibers of rational maps and elimination matrices: an application oriented approach

Parameterized algebraic curves and surfaces are widely used in geometric modeling and their manipulation is an important task in the processing of geometric models. In particular, the determination of the intersection loci between points, pieces of parameterized algebraic curves and pieces of algebraic surfaces is a key problem in this context. In this paper, we survey recent methods based on syzygies and blowup algebras for computing the image and the finite fibers of a curve or surface parameterization, more generally of a rational map. Conceptually, the main idea is to use elimination matrices, mainly built from syzygies, as representations of rational maps and to extract geometric informations from them. The construction and main properties of these matrices are first reviewed and then illustrated through several settings, each of them highlighting a particular feature of this approach that combines tools from commutative algebra, algebraic geometric and elimination theory.

math.AC↗

arXiv2020

Determinantal tensor product surfaces and the method of moving quadrics

A tensor product surface $\mathscr{S}$ is an algebraic surface that is defined as the closure of the image of a rational map $ϕ$ from $\mathbb{P}^1\times \mathbb{P}^1$ to $\mathbb{P}^3$. We provide new determinantal representations of $\mathscr{S}$ under the assumptions that $ϕ$ is generically injective and its base points are finitely many and locally complete intersections. These determinantal representations are matrices that are built from the coefficients of linear relations (syzygies) and quadratic relations of the bihomogeneous polynomials defining $ϕ$. Our approach relies on a formalization and generalization of the method of moving quadrics introduced and studied by David Cox and his co-authors.

math.AG↗

arXiv2020

Fibers of multi-graded rational maps and orthogonal projection onto rational surfaces

We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite fibers of a generically finite dominant rational map: a congruence of normal lines to the rational surface. Then, an in-depth study of certain syzygy modules associated to such a congruence is presented and applied to build elimination matrices that provide universal representations of its finite fibers, under some genericity assumptions. These matrices depend linearly in the variables of the three dimensional space. They can be pre-computed so that the orthogonal projections of points are approximately computed by means of fast and robust numerical linear algebra calculations.

math.AC↗

arXiv2020

Degree and birationality of multi-graded rational maps

We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian dual criterion to the multi-graded setting. Our approach is based on the study of blow-up algebras, including syzygies, of the ideal generated by the defining polynomials of the rational map. A key ingredient is a new algebra that we call the "saturated special fiber ring", which turns out to be a fundamental tool to analyze the degree of a rational map. We also provide a very effective birationality criterion and a complete description of the equations of the associated Rees algebra of a particular class of plane rational maps.

math.AG↗

arXiv2020

Freeness and invariants of rational plane curves

Given a parameterization $ϕ$ of a rational plane curve C, we study some invariants of C via $ϕ$. We first focus on the characterization of rational cuspidal curves, in particular we establish a relation between the discriminant of the pull-back of a line via $ϕ$, the dual curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via $ϕ$, we prove that the (nearly) freeness of a rational curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a rational curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C.

math.AG↗

arXiv2019

The geometry of the flex locus of a hypersurface

We give a formula in terms of multidimensional resultants for an equation for the flex locus of a projective hypersurface, generalizing a classical result of Salmon for surfaces. Using this formula, we compute the dimension of this flex locus, and an upper bound for the degree of its defining equations. We also show that, when the hypersurface is generic, this bound is reached, and that the generic flex line is unique and has the expected order of contact with the hypersurface.

math.AG↗

arXiv2017

Discriminants of complete intersection space curves

In this paper, we develop a new approach to the discrimi-nant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discrimi-nant without ambiguity and over any commutative ring, in particular in any characteristic. This formula also provides a new method for evaluating and computing this discrimi-nant efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we obtain new properties and computational rules such as the covariance and the invariance formulas. Finally, we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. Actually, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.

cs.SC↗

arXiv2016

Extraction of cylinders and cones from minimal point sets

We propose new algebraic methods for extracting cylinders and cones from minimal point sets, including oriented points. More precisely, we are interested in computing efficiently cylinders through a set of three points, one of them being oriented, or through a set of five simple points. We are also interested in computing efficiently cones through a set of two oriented points, through a set of four points, one of them being oriented, or through a set of six points. For these different interpolation problems, we give optimal bounds on the number of solutions. Moreover, we describe algebraic methods targeted to solve these problems efficiently.

cs.CG↗