Double-exponential lower bound for the degree of any system of generators of a polynomial prime ideal

Let $A$ be a polynomial ring in $n+1$ variables over an arbitrary infinite field $k$. It is proved that for all sufficiently large $n$ and $d$ there is a homogeneous prime ideal ${\mathfrak p}\subset A$ satisfying the following conditions. The ideal ${\mathfrak p}$ corresponds to a component, defined over $k$ and irreducible over $\overline{k}$, of a projective algebraic variety given by a system of homogeneous polynomial equations with polynomials in $A$ 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 a slightly less strong result is obtained.