**Luogo:** Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula Seminari "-1"**Ore** 16.00

**Relatore:**Fernando Galvàn Josè Francisco (Departamento de Algebra, Facultad de Ciencias Matematicas, Universidad ́Complutense de Madrid)

**Abstract:**

Let f := (f1, . . . , fm) : R n → R m be a map. We say that f is polynomial if its components fk are polynomials. The map f is regular if its components can be represented as quotients fk = gk hk of two polynomials gk, hk such that hk never vanishes on R n . More generally, the map f is Nash if each component fk is a Nash function, that is, an analytic function whose graph is a semialgebraic set. Recall that a subset S ⊂ R n is semialgebraic if it has a description as a finite boolean combination of polynomial equalities and inequalities. By Tarski-Seidenberg’s principle the image of an either polynomial, regular or Nash map is a semialgebraic set. In 1990 Oberwolfach reelle algebraische Geometrie week, the second author proposed a kind of converse problem: To characterize the semialgebraic sets in R m that are either polynomial or regular images of some R n . In the same period Shiota formulated a conjecture that characterizes Nash images of R n , that has been recently solved by the first author. In this seminar we collect our main contributions (jointly with J.M. Gamboa and C. Ueno) to these problems. We have approached them in three directions: (i) Explicit construction of polynomial and regular maps whose images are the members of large families of semialgebraic sets whose boundaries are piecewise linear. (ii) To find obstructions to be polynomial/regular images of R n . (iii) To prove Shiota’s conjecture and some relevant consequences.

**Referente: **Riccardo Ghiloni