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Register a new userVirtual rigid motives of semi-algebraic sets
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Authors
Arthur Forey
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Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend the morphism sending the class of an algebraic variety over $K$ to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdans motivic integration and Ayoubs equivalence between motives of rigid analytic varieties over $K$ and quasi-unipotent motives over $k$ ; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.
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Given any arbitrary semi-algebraic set $X$, any two points in $X$ may be joined by a piecewise $C^2$ path $gamma$ of shortest length. Suppose $mathcal{A}$ is a semi-algebraic stratification of $X$ such that each component of $gamma cap mathcal{A}$ is either a singleton or a real analytic geodesic segment in $mathcal{A}$, the question is whether $gamma cap mathcal{A}$ has at most finitely many such components. This paper gives a semi-algebraic stratification, in particular a cell decomposition, of a real semi-algebraic set in the plane whose open cells have this finiteness property. This provides insights for high dimensional stratifications of semi-algebraic sets in connection with geodesics.
We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the corresponding intermediate Jacobian J^2(X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibered by surfaces with algebraic H^2. This gives another new examples of three-dimensional varieties whose motives are finite-dimensional.
In a number of recent works [6, 7] the authors have introduced and studied a functor $mathcal{F}_k$ which associates to each loose graph $Gamma$ -which is similar to a graph, but where edges with $0$ or $1$ vertex are allowed - a $k$-scheme, such that $mathcal{F}_k(Gamma)$ is largely controlled by the combinatorics of $Gamma$. Here, $k$ is a field, and we allow $k$ to be $mathbb{F}_1$, the field with one element. For each finite prime field $mathbb{F}_p$, it is noted in [6] that any $mathcal{F}_k(Gamma)$ is polynomial-count, and the polynomial is independent of the choice of the field. In this note, we show that for each $k$, the class of $mathcal{F}_k(Gamma)$ in the Grothendieck ring $K_0(texttt{Sch}_k)$ is contained in $mathbb{Z}[mathbb{L}]$, the integral subring generated by the virtual Lefschetz motive.
Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected compact component of a real (semi)-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the $n=4$ case. We also apply our method to design an efficient algorithm to compute the real dimension of (semi)-algebraic sets, the original motivation for this research.
Let $Ssubset R^n$ be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. We design an algorithm which takes as input a polynomial system defining $S$ and an integer $pgeq 0$ and returns the $n$-dimensional volume of $S$ at absolute precision $2^{-p}$.Our algorithm relies on the relationship between volumes of semi-algebraic sets and periods of rational integrals. It makes use of algorithms computing the Picard-Fuchs differential equation of appropriate periods, properties of critical points, and high-precision numerical integration of differential equations.The algorithm runs in essentially linear time with respect to~$p$. This improves upon the previous exponential bounds obtained by Monte-Carlo or moment-based methods. Assuming a conjecture of Dimca, the arithmetic cost of the algebraic subroutines for computing Picard-Fuchs equations and critical points is singly exponential in $n$ and polynomial in the maximum degree of the input.
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