[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