[Author]
A. L. Chistov
[Title]
A Bound for the Degree of a System of Equations Giving the Variety of
Reducible Polynomials
[AMS Subj-class]
14Q15 Higher-dimensional varieties
14M99 Algebraic geometry
12Y05 Computational aspects of field theory and polynomials
12E05 Polynomials (irreducibility, etc.)
13P05 Polynomials, factorization
[Abstract]
Consider the affine space ${\mathbb A}^N(\overline{K})$ of homogeneous
polynomials of degree $d$ in $n+1$ variables with coefficients from an
algebraic closure $\overline{K}$ of a field $K$ of arbitrary
characteristic, so $N={n+d \choose n}$.
We prove that the variety of all reducible polynomials from this affine
space can be given by a system of polynomial equations of degree less
than $56d^7$ in $N$ variables.
Using this result we formulate an effective version of the first
Bertini theorem for the case of a hypersurface.
[Keywords]
absolute irreducibility, lattices, the Bertini theorem
[Comments]
LaTeX, English, 18 pp.
To appear in: "Алгебра и анализ" No.3, 2012, in Russian.
In the second version of this preprint the upper bound $84d^7$ is
improved. It is replaced by $56d^7$.
[Contact e-mail]
alch@pdmi.ras.ru