Quantitative Singularity Theory for Random Polynomials

Abstract

Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-$W$ singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-$W$ singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset $W$ of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-$W$ singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function $p$ we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center $p$, in which the $W$-singularity structure is constant.

On the topology of random real complete intersections

“…This is the content of Corollary 5.3 which will use the estimates on the distance to the real discriminant Σ d proved in Section 3.2. In the case of Kostlan polynomials similar results can be found in [5,Proposition 3] and[3, Theorem 7]. Let s ∈ RH 0 (X, ⊕ m i=1 L d i ) \ Σ d be a real section (see Definition 3.1 for the definition of the real discriminant Σ d ).…”

“…In our case, Theorem 1.4 is a consequence of the low degree approximation property given by Theorem 1.2. This has already been observed first in the case of Kostlan complete intersections in P n in [3,5] and then in the case of real hypersurfaces in a real algebraic variety in [2].…”

Functional norms, condition numbers and numerical algorithms in algebraic geometry

“…We finish with the following theorem, similar in flavour to [28, Proposition 3] and [11,Theorem 7], where it was shown that the distance of a polynomial tuple to polynomial tuples with singularities bounds the distance of this polynomial to C 1 -functions with singularities.…”

Kac-Rice formula for transverse intersections

“…The meaning of "singular point" depends on the situation, but in general it is a point p ∈ M where the section satisfies some condition involving its derivatives. A general model for that (the same proposed in [25] and [7]) is to consider a subset W ⊂ J r E of the bundle of r jets (if the derivatives involved are of order less than r) of sections of E (see [17] for a definition of the space of jets) and call singular points of class W those points p ∈ M such that the r th jet of X at p belongs to W . Examples are:…”