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Daniel Perrucci

Publications and source records attributed to Daniel Perrucci.

18 recordsLinked to original sources
arXiv2023

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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arXiv2023

Rational certificates of non-negativity on semialgebraic subsets of cylinders

Let $g_1,\dots, g_s \in \mathbb{R}[X_1,\dots, X_n,Y]$ and $S = \{(\bar{x},y)\in \mathbb{R}^{n+1} \mid g_1(\bar{x},y) \ge 0, \dots, g_s(\bar{x}, y) \ge 0\}$ be a non-empty, possibly unbounded, subset of a cylinder in $\mathbb{R}^{n+1}$. Let $f \in \mathbb{R}[X_1, \dots, X_n, Y]$ be a polynomial which is positive on $S$. We prove that, under certain additional assumptions, for any non-constant polynomial $q \in \mathbb{R}[Y]$ which is positive on $\mathbb{R}$, there is a certificate of the non-negativity of $f$ on $S$ given by a rational function having as numerator a polynomial in the quadratic module generated by $g_1, \dots, g_s$ and as denominator a power of $q$.

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arXiv2022

Topology of real multi-affine hypersurfaces and a homological stability property

Let $\mathrm{R}$ be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in $\mathrm{R}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by $2^{d-1}$. This bound is sharp and is independent of $n$ (as opposed to the classical bound of $d(2d -1)^{n-1}$ on the Betti numbers of hypersurfaces defined by arbitrary polynomials of degree $d$ in $\mathrm{R}^n$ due to Petrovski{\u\i} and Ole{\u\i}nik, Thom and Milnor). Moreover, we show there exists $c > 1$, such that given a sequence $(B_n)_{n >0}$ where $B_n$ is a closed ball in $\mathrm{R}^n$ of positive radious, there exist hypersurfaces $(V_n)_{n_>0}$ defined by symmetric multi-affine polynomials of degree $4$, such that $\sum_{i \leq 5} b_i(V_n \cap B_n) > c^n$, where $b_i(\cdot)$ denotes the $i$-th Betti number with rational coeffcients. Finally, as an application of the main result of the paper we verify a representational stability conjecture due to Basu and Riener on the cohomology modules of symmetric real algebraic sets for a new and much larger class of symmetric real algebraic sets than known before.

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arXiv2021

A few more extensions of Putinar's Positivstellensatz to non-compact sets

We extend previous results about Putinar's Positivstellensatz for cylinders of type $S \times {\mathbb R}$ to sets of type $S \times {\mathbb R}^r$ in some special cases taking into account $r$ and the degree of the polynomial with respect to the variables moving in ${\mathbb R}^r$ (this is to say, in the non-bounded directions). These special cases are in correspondence with the ones where the equality between the cone of non-negative polynomials and the cone of sums of squares holds. Degree bounds are provided.

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arXiv2019

On sum of squares certificates of non-negativity on a strip

A well-known result of Murray Marshall states that every $f \in \mathbb{R} [X,Y]$ non-negative on the strip $[0,1] \times \mathbb{R}$ can be written as $f= \sigma_0 + \sigma_1 X(1-X)$ with $\sigma_0, \sigma_1$ sums of squares in $\mathbb{R} [X,Y]$. In this work, we present a few results concerning this representation in particular cases. First, under the assumption ${\rm deg}_Y f \leq 2$, by characterizing the extreme rays of a suitable cone, we obtain a degree bound for each term. Then, we consider the case of $f$ positive on $[0,1] \times \mathbb{R}$ and non-vanishing at infinity, and we show again a degree bound for each term, coming from a constructive method to obtain the sum of squares representation. Finally, we show that this constructive method also works in the case of $f$ having only a finite number of zeros, all of them lying on the boundary of the strip, and such that $\frac{\partial f}{\partial X}$ does not vanish at any of them.

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arXiv2018

A version of Putinar's Positivstellensatz for cylinders

We prove that, under some additional assumption, Putinar's Positivstellensatz holds on cylinders of type $S \times {\mathbb R}$ with $S = \{x \in {\mathbb R}^n | g_1(x) \ge 0, ..., g_s(x) \ge 0\}$ such that the quadratic module generated by $g_1, ..., g_s$ in ${\mathbb R}[X_1, ..., X_n]$ is archimedean, and we provide a degree bound for the representation of a polynomial $f \in {\mathbb R}[X_1, ..., X_n, Y]$ which is positive on $S \times {\mathbb R}$ as an explicit element of the quadratic module generated by $g_1, ..., g_s$ in ${\mathbb R}[X_1, ..., X_n, Y]$. We also include an example to show that an additional assumption is necessary for Putinar's Positivstellensatz to hold on cylinders of this type.

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arXiv2018

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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arXiv2016

On the Davenport-Mahler bound

We prove that the Davenport-Mahler bound holds for arbitrary graphs with vertices on the set of roots of a given univariate polynomial with complex coefficients.

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arXiv2014

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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arXiv2013

Zero-nonzero and real-nonreal sign determination

We consider first the zero-nonzero determination problem, which consists in determining the list of zero-nonzero conditions realized by a finite list of polynomials on a finite set Z included in C^k with C an algebraic closed field. We describe an algorithm to solve the zero-nonzero determination problem and we perform its bit complexity analysis. This algorithm, which is in many ways an adaptation of the methods used to solve the more classical sign determination problem, presents also new ideas which can be used to improve sign determination. Then, we consider the real-nonreal sign determination problem, which deals with both the sign determination and the zero-nonzero determination problem. We describe an algorithm to solve the real-nonreal sign determination problem, we perform its bit complexity analysis and we discuss this problem in a parametric context.

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arXiv2011

On the minimum of a polynomial function on a basic closed semialgebraic set and applications

We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is not zero. As an application, we obtain a lower bound for the separation of two disjoint connected components of basic closed semialgebraic sets, when at least one of them is compact.

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arXiv2009

Linear Solving for Sign Determination

We give a specific method to solve with quadratic complexity the linear systems arising in known algorithms to deal with the sign determination problem. In particular, this enable us to improve the complexity bound for sign determination in the univariate case.

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arXiv2009

On the minimum of a positive polynomial over the standard simplex

We present a new positive lower bound for the minimum value taken by a polynomial P with integer coefficients in k variables over the standard simplex of R^k, assuming that P is positive on the simplex. This bound depends only on the number of variables, the degree and the bitsize of the coefficients of P and improves all previous bounds for arbitrary polynomials which are positive over the simplex.

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arXiv2008

On sign conditions over real multivariate polynomials

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions over the polynomials. In addition, we extend these results to the case of closed sign conditions over an arbitrary family of real multivariate polynomials. The complexity bounds for these procedures improve the known ones.

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arXiv2005

Some Bounds for the Number of Components of Real Zero Sets of Sparse Polynomials

We prove that the zero set of a 4-nomial in n variables in the positive orthant has at most three connected components. This bound, which does not depend on the degree of the polynomial, not only improves the best previously known bound (which was 10) but is optimal as well. In the general case, we prove that the number of connected components of the zero set of an m-nomial in n variables in the positive orthant is lower than or equal to (n+1)^{m-1}2^{1 + (m - 1)(m - 2)/2}, improving slightly the known bounds. Finally, we show that for generic exponents, the number of non-compact connected components of the zero set of a 5-nomial in three variables in the positive octant is at most 12. This strongly improves the best previously known bound, which was 10384. All the bounds obtained in this paper continue to hold for real exponents.

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