A bound for the degree of a system of equations determining the variety of reducible polynomials (Q2849092)

The author of this paper faces from a computational point of view the problem of verifying if a polynomial \(f\) in the ring \(K[x_1,\ldots,x_n]\) is irreducible, over the algebraic closure \(\bar K\) of the field \(K\).NEWLINENEWLINEBy several very technical lemmas, the construction of a particular polynomial \(A_f\) in the coefficients of \(f\) is introduced, as the product between two suitable resultants. Here, the polynomial \(f\) is irreducible if and only if the product \(A_f\) is non null (Lemma 10, in Section 3), in analogy with the fact that a polynomial in one variable is square-free if and only if its discriminant is non-null.NEWLINENEWLINEThe role of \(A_f\) in the irreducibility of \(f\) is described first in the case \(f\) has only two variables (Lemma 6, in Section 2), and then is lifted to the case of more than two variables.NEWLINENEWLINEThe author applies the study of \(A_f\) to obtain the two results presented in the Introduction as Theorem 1 and Theorem 2, respectively.NEWLINENEWLINETheorem 1 states that the reducible hypersurfaces of a given degree \(d\) form a Zariski closed subset in the set of all the hypersurfaces of degree \(d\) in a fixed affine (resp. projective) space and gives an upper bound \(<56d^{7}\) for the degree of the equations defining this closed subset. Whereas the first statement is a particular case of a more general known result in Hilbert schemes (for example, see Théorème (12.2.1) in [\textit{A.~Grothendieck}, Éléments de géométrie algébrique. {IV}. Étude locale des schémas et des morphismes de schémas {IV}, Publ. Math., Inst. Hautes Étud. Sci. 32, 1--361 (1967; Zbl 0153.22301)], the second one is due to the construction of \(A_f\) (see Lemma 11 in Section 3) and is the main result of this paper, as highlighted in its title.NEWLINENEWLINEGiven a polynomial \(f\) that is irreducible over \(\bar K\), Theorem 2 provides some conditions under which a suitable class of linear polynomials determines irreducible hyperplane sections of the hypersurface defined by \(f\). A procedure to construct such linear polynomials is also presented.NEWLINENEWLINEThis paper is one of a series of papers in which the author deals with the interesting topic of how to make classical results in Algebraic Geometry be effective. Recently, a correction to this paper has been published by the author in the same journal.