[Author]
A. L. Chistov
[Title]
Double-exponential lower bound for the degree of any system of
generators of a polynomial prime ideal
[AMS Subj-class]
14Q15 Higher-dimensional varieties
[Abstract]
Consider a polynomial ring $A$ in $n+1$ variables over an arbitrary
infinite field $k$. We prove that for all sufficiently big $n$ and
$d$ there is a homogeneous prime ideal ${\mathfrak p}\subset A$
satisfying the following conditions. The ideal ${\mathfrak p}$
corresponds to a defined over $k$ and irreducible over
$\overline{k}$ component of a projective algebraic variety given by
a system of homogeneous polynomial equations with polynomials from
$A$ of degrees less than $d$. Any system of generators of ${\mathfrak
p}$ contains a polynomial of degree at least $d^{2^{cn}}$ for an
absolute constant $c>0$ which can be computed efficiently.
This solves an important old problem in effective algebraic geometry.
For the case of finite fields we obtain a slightly less strong result.
[Comments]
LaTeX, English (21 pp.) & Russian (23 pp.)
[Contact e-mail]
alch@pdmi.ras.ru