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Marie-Françoise Roy

Publications and source records attributed to Marie-Françoise Roy.

12 recordsLinked to original sources
arXiv2025

Dynamical method in algebra: Effective Nullstellensätze

We give a general method for producing various effective Null and Positivstellensätze, and getting new Positivstellensätze in algebraically closed valued fields and ordered groups. These various effective Nullstellensätze produce algebraic identities certifying that some geometric conditions cannot be simultaneously satisfied. We produce also constructive versions of abstract classical results of algebra based on Zorn's lemma in several cases where such constructive version did not exist. For example, the fact that a real field can be totally ordered, or the fact that a field can be embedded in an algebraically closed field. Our results are based on the concepts we develop of dynamical proofs and simultaneous collapse.

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arXiv2024

Algebraic Winding Numbers

In this paper, we propose a new algebraic winding number and prove that it computes the number of complex roots of a polynomial in a rectangle, including roots on edges or vertices with appropriate counting. The definition makes sense for the algebraic closure C = R[i] of a real closed field R, and the root counting result also holds in this case. We study in detail the properties of the algebraic winding number defined in [3] with respect to complex root counting in rectangles. We extend both winding numbers to rational functions, obtaining then algebraic versions of the argument principle for rectangles.

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arXiv2021

Bounds for polynomials on algebraic numbers and application to curve topology

Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total degree $d$ with integer coefficients of bitsize less than $τ$, and let $V_{\mathbb{R}} (P) := \{ (x,y) \in \mathbb{R}^2, P (x,y) = 0 \}$ be the real planar algebraic curve implicitly defined as the vanishing set of $P$. We give a deterministic and certified algorithm to compute the topology of $V_{\mathbb{R}} (P)$ in terms of a straight-line planar graph $\mathcal{G}$ that is isotopic to $V_{\mathbb{R}} (P)$. Our analysis yields the upper bound $\tilde O (d^5 τ+ d^6)$ on the bit complexity of our algorithm, which matches the current record bound for the problem of computing the topology of a planar algebraic curve However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and the returned graph $\mathcal{G}$ yields the cylindrical algebraic decomposition information of the curve. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of such polynomials that actually exploit this amortization. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all critical points.

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arXiv2021

Quantitative Curve Selection Lemma

We prove a quantitative version of the curve selection lemma. Denoting by $s,d,k$ a bound on the number, the degree and the number of variables of the polynomials describing a semi-algebraic set $S$ and a point $x$ in $\bar S$, we find a semi-algebraic path starting at $x$ and entering in $S$ with a description of degree $(O(d)^{3k+3},O(d)^{k})$ (using a precise definition of the description of a semi-algebraic path and its degree given in the paper). As a consequence, we prove that there exists a semi-algebraic path starting at $x$ and entering in $S$, such that the degree of the Zariski closure of the image of this path is bounded by $O(d)^{4k+3}$, improving a result of Jelonek and Kurdyka. We also give an algorithm for describing the real isolated points of $S$ whose complexity is bounded by $s^{2 k+1}d^{O(k)}$ improving a result of Le, Safey el Din, and de Wolff.

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arXiv2019

A data analysis of women's trails among ICM speakers

The International Congress of Mathematicians (ICM), inaugurated in 1897, is the greatest effort of the mathematical community to strengthen international communication and connections across all mathematical fields. Meetings of the ICM have historically hosted some of the most prominent mathematicians of their time. Receiving an invitation to present a talk at an ICM signals the high international reputation of the recipient, and is akin to entering a `hall of fame for mathematics'. Women mathematicians attended the ICMs from the start. With the invitation of Laura Pisati to present a lecture in 1908 in Rome and the plenary talk of Emmy Noether in 1932 in Zurich, they entered the grand international stage of their field. At the congress in 2014 in Seoul, Maryam Mirzakhani became the first woman to be awarded the Fields Medal, the most prestigious award in mathematics. In this article, we dive into assorted data sources to follow the footprints of women among the ICM invited speakers, analyzing their demographics and topic distributions, and providing glimpses into their diverse biographies.

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arXiv2019

Sylvester double sums, subresultants and symmetric multivariate Hermite interpolation

Sylvester doubles sums, introduced first by Sylvester are symmetric expressions of the roots of two polynomials. Sylvester's definition of double sums makes no sense in the presence of multiple roots, since the definition involves denominators that vanish when there are multiple roots. The aim of this paper is to give a new definition of Sylvester double sums making sense in the presence of multiple roots, which coincides with the definition by Sylvester in the case of simple roots, to prove that double sums indexed by $(k,\ell)$ are equal up to a constant if they share the same value for $k+\ell$, as well a proof of the relationship between double sums and subresultants, i.e. that they are equal up to a constant. In the simple root case, proofs of these properties are already known. The more general proofs given here are using generalized Vandermonde determinants and symmetric multivariate Hermite interpolation as well as an induction on the length of the remainder sequence of $P$ and $Q$.

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arXiv2019

Quantitative Fundamental Theorem of Algebra

Using subresultants, we modify a recent real-algebraic proof due to Eisermann of the Fundamental Theorem of Algebra ([FTA]) to obtain the following quantitative information: in order to prove the [FTA] for polynomials of degree $d$, the Intermediate Value Theorem ([IVT]) is requested to hold for real polynomials of degree at most $d^2$. We also explain that the classical proof due to Laplace requires [IVT] for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.

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arXiv2017

An elementary recursive bound for effective Positivstellensatz and Hilbert 17-th problem

We prove elementary recursive bounds in the degrees for Positivstellensatz and Hilbert 17-th problem, which is the expression of a nonnegative polynomial as a sum of squares of rational functions. We obtain a tower of five exponentials. A precise bound in terms of the number and degree of the polynomials and their number of variables is provided in the paper.

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arXiv2014

A baby step-giant step roadmap algorithm for general algebraic sets

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We give an algorithm that takes as input a polynomial $Q \in \mathrm{D}[X_1,\ldots,X_k]$, and computes a description of a roadmap of the set of zeros, $\mathrm{Zer}(Q,\mathrm{R}^k)$, of $Q$ in $\mathrm{R}^k$. The complexity of the algorithm, measured by the number of arithmetic operations in the ordered domain $\mathrm{D}$, is bounded by $d^{O(k \sqrt{k})}$, where $d = \mathrm{deg}(Q)\ge 2$. As a consequence, there exist algorithms for computing the number of semi-algebraically connected components of a real algebraic set, $\mathrm{Zer}(Q,\mathrm{R}^k)$, whose complexity is also bounded by $d^{O(k \sqrt{k})}$, where $d = \mathrm{deg}(Q)\ge 2$. The best previously known algorithm for constructing a roadmap of a real algebraic subset of $\mathrm{R}^k$ defined by a polynomial of degree $d$ has complexity $d^{O(k^2)}$.

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arXiv2008

Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials

Let $\R$ be a real closed field, $ {\mathcal Q} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k], $ with $ °_{Y}(Q) \leq 2, °_{X}(Q) \leq d, Q \in {\mathcal Q}, #({\mathcal Q})=m$, and $ {\mathcal P} \subset \R[X_1,...,X_k] $ with $°_{X}(P) \leq d, P \in {\mathcal P}, #({\mathcal P})=s$. Let $S \subset \R^{\ell+k}$ be a semi-algebraic set defined by a Boolean formula without negations, with atoms $P=0, P \geq 0, P \leq 0, P \in {\mathcal P} \cup {\mathcal Q}$. We describe an algorithm for computing the the Betti numbers of $S$. The complexity of the algorithm is bounded by $(\ell s m d)^{2^{O(m+k)}}$. The complexity of the algorithm interpolates between the doubly exponential time bounds for the known algorithms in the general case, and the polynomial complexity in case of semi-algebraic sets defined by few quadratic inequalities known previously. Moreover, for fixed $m$ and $k$ this algorithm has polynomial time complexity in the remaining parameters.

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