Friday, April 17
on-line on zoom
Meeting ID: 931 2534 5558
Passcode: 983090
- 9:45-10:35 Riccardo Ghiloni (Università di Trento) The Nash-Tognoli theorem over the rationals and its version for isolated singularities
Abstract: The Nash-Tognoli theorem asserts that every compact smooth manifold \(M\) of dimension \(d\) is smoothly diffeomorphic to a nonsingular algebraic subset \(X\) of \(\mathbb{R}^{2d+1}\), a so-called algebraic model of \(M\). Since the subset \(X\) of \(\mathbb{R}^n\) is algebraic and nonsingular, it can be described both globally and locally by a finite number of polynomials with real coefficients, that is, there exist polynomials \(p_1,\ldots,p_n\) in \(\mathbb{R}[x_1,...,x_{2d+1}]\) for a certain positive natural number \(n\) such that:
- \(X\) is the locus of the zeros of the polynomials \(p_i\), that is, \(X=\{x\in\mathbb{R}^{2d+1}:p_1(x)=0,\ldots,p_n(x)=0\}\) and
- locally at every point \(y\) of \(X\), the polynomials \(p_i\) describe the differential structure of \(X\) via the Implicit Function Theorem, that is, for every \(y\in X\), there exist a neighborhood \(U\) of \(y\) in \(\mathbb{R}^{2d+1}\) and \(d+1\) indices \(i_1,\ldots,i_{d+1}\) extracted from \(\{1,..,n\}\) (that depend on \(y\)) such that \(X\cap U=\{x\in U:p_{i_1}(x)=0,\ldots,p_{i_{d+1}}(x)=0\}\) and the gradients of the polynomials \(p_{i_1},...,p_{i_{d+1}}\) evaluated at \(y\) are linearly independent.
The aim of the talk is to present this Nash-Tognoli theorem over the rationals and also a version of it for real algebraic sets with isolated singularities.
The previous results are based in part on a new version of real algebraic geometry that I recently developed together with José F. Fernando in [2].
References:
[1] Riccardo Ghiloni and Enrico Savi, The Nash-Tognoli theorem over the rationals and its version for isolated singularities, arXiv:2302.04142v2
[2] José F. Fernando and Riccardo Ghiloni, Subfield-algebraic geometry, arXiv:2512.08975
- 10:45-11:35 Antonio Carbone (Università di Ferrara) Resolution of closed semialgebraic sets and applications
Abstract: Bierstone and Parusiński studied the desingularization of \(d\)-dimensional closed subanalytic sets and in particular of \(d\)-dimensional closed semialgebraic sets. Their procedure preserves the number of \(d\)-dimensional components connected by analytic paths of the involved closed semialgebraic sets, so they have a good behaviour for pure dimensional closed semialgebraic sets. If the involved \(d\)-dimensional closed semialgebraic set is not pure dimensional, some components connected by analytic paths of smaller dimension could be dropped during the desingularization process. For instance, Whitney's umbrella \(W:=\{{\tt y}^2{\tt z}-{\tt x}^2=0\}\subset\mathbb{R}^3\) has two components connected by analytic paths (one of dimension \(2\) and the other of dimension \(1\)), whereas its desingularization has only one component connected by analytic paths, that has dimension \(2\). The obtained models in the desingularization process, that we call closed chessboard sets, are the closures of (finite) unions of connected components of the complements of normal-crossings divisors of non-singular real algebraic sets.
In this seminar we start by presenting the resolution of \(d\)-dimensional closed chessboard sets \(\mathcal{S}\) using Nash manifolds with corners \(\mathcal{Q}\) with the same number of connected components as \(\mathcal{S}\) (or equivalently the same number of irreducible components). Next, we present the Nash double \(D(\mathcal{Q})\) of a Nash manifold with corners \(\mathcal{Q}\), which is the analogous of the Nash double of a Nash manifold with smooth boundary, but takes into account the peculiarities of the boundary of \(\mathcal{Q}\). Combining the previous results with Bierstone and Parusiński's desingularization (together with some algebraization techniques) we obtain that: If \(\mathcal{S}\) is a \(d\)-dimensional closed semialgebraic set connected by analytic paths, then there exists a \(d\)-dimennsional non-singular irreducible real algebraic set \(X\), a proper surjective polynomial map \(f:X\to \mathcal{S}\) and a closed semialgebraic subset \(\mathcal{R}\subset \mathcal{S}\) of dimension \(<d\) such that \(X\setminus f^{-1}(\mathcal{R})\) and \(\mathcal{S}\setminus \mathcal{R}\) are Nash manifolds of the same dimension \(d\) and the restriction of \(f\) to \(X\setminus f^{-1}(\mathcal{R})\) is a Nash covering map whose fibers are finite and have constant cardinality. Finally, we present several applications of our results:
- A complete characterisation of the compact semialgebraic sets that are images of the unitary closed ball under Nash maps.
- A compactification result for Nash manifolds with corners by compact Nash manifolds with corners.
- A result on Nash approximation of continuous semialgebraic maps whose target space are Nash manifolds with corners.