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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 p ⊂ A satisfying the following conditions. The ideal p corresponds to a component, defined over k and irreducible over s 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 p contains a polynomial of degree at least d2 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.

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