Friday, April 17
on-line on zoom
- 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
- 11:45-12:35 Antonio Carbone (Università di Ferrara)TBA