A Szemeredi-Trotter type theorem in R4\mathbb{R}^4

We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\leq n^{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemer\'edi-Trotter type theorem for 2--planes in $\mathbb{R}^4$, provided $m\leq n$ and the planes intersect transversely. As a further special case, we obtain a Szemer\'edi-Trotter type theorem for complex lines in $\mathbb{C}^2$ with no restrictions on $m$ and $n$ (this theorem was originally proved by T\'oth using a different method). As a third special case, we obtain a Szemer\'edi-Trotter type theorem for complex unit circles in $\mathbb{C}^2$. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.Comment: 50 pages. V3: final version. To appear in Discrete and Computational Geometr

Numeric and Certified Isolation of the Singularities of the Projection of a Smooth Space Curve

International audienceLet CP ∩Q be a smooth real analytic curve embedded in R 3 , defined as the solutions of real analytic equations of the form P (x, y, z) = Q(x, y, z) = 0 or P (x, y, z) = ∂P ∂z = 0. Our main objective is to describe its projection C onto the (x, y)-plane. In general, the curve C is not a regular submanifold of R 2 and describing it requires to isolate the points of its singularity locus Σ. After describing the types of singularities that can arise under some assumptions on P and Q, we present a new method to isolate the points of Σ. We experimented our method on pairs of independent random polynomials (P, Q) and on pairs of random polynomials of the form (P, ∂P ∂z) and got promising results

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

International audienceEffective quantifier elimination procedures for first-order theories provide a powerful tool for genericallysolving a wide range of problems based on logical specifications. In contrast to general first-order provers, quantifierelimination procedures are based on a fixed set of admissible logical symbolswith an implicitly fixed semantics. Thisadmits the use of sub-algorithms from symbolic computation. We are going to focus on quantifier elimination forthe reals and its applications giving examples from geometry, verification, and the life sciences. Beyond quantifierelimination we are going to discuss recent results with a subtropical procedure for an existential fragment of thereals. This incomplete decision procedure has been successfully applied to the analysis of reaction systems inchemistry and in the life sciences

An Elementary Approach to Subresultants Theory

In this paper we give an elementary approach to univariate polynomial subresultants theory. Most of the known results of subresultants are recovered, some with more precision, without using Euclidean divisions or existence of roots for univariate polynomials. The main contributions of this paper are not new results on subresultants, but rather extensions of the main results over integral rings to arbitrary commutative rings

An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve

The computation of the topological shape of a real algebraic plane curve is usually driven by the study of the behavior of the curve around its critical points (which includes also the singular points). In this paper we present a new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms known. This is due to the avoiding, through a sufficiently good change of coordinates, of real root computations on polynomials with coefficients in a simple real algebraic extension of $\mathbb{Q}$ to deal with the critical points of the considered curve. In fact, one of the main features of this algorithm is that its complexity is dominated by the characterization of the real roots of the discriminant of the polynomial defining the considered curve

Improved algorithms for linear complementarity problems arising from collision response

In this paper we give algorithms for solving linear complementarity problems for $\mathcal{P}$-matrices and symmetric positive semidefinite matrices. Our approach of the problem turns out to be an improvement and a more precise formulation of Baraff’s method for problems arising from collision response. The theorems that prove the correctness of our algorithm can also be used to prove the correctness of Baraff’s algorithm. An important feature of the method we present lies in its validity for arbitrary real closed fields, thus it is well suited to handle, at least locally, parametric linear complementarity problems. This article presents the theoretical principles of the algorithms and gives detailed pseudo-code descriptions of them

Birational properties of the gap subresultant varieties

In this paper we address the problem of understanding the gaps that may occur in the subresultant sequence of two polynomials.We define the gap subresultant varieties and prove that they are rational and have the expected dimension. We also give explicitly their corresponding prime ideals