On zeros of bounded degree of systems of homogeneous polynomial equations

Georg Eulering and Martin Krüskemper

Vol. 180 (1997), No. 1, 1–5

DOI: 10.2140/pjm.1997.180.1

Abstract

Let F be a finite or algebraically closed field and R = F[T₁,…,T_s], the polynomial ring in T₁,…,T_s over F. Then by Tsen-Lang, any system of homogeneous polynomials f₁(X),… , f_r(X) ∈ R[X] of degree d, where = (X₁,…,X_n), has a non-trivial common zero in Rⁿ provided the number of variables n is sufficiently large. In this note we want to give an effective bound B such that there exists a zero 0≠(a₁,…,a_n) ∈ Rⁿ with max{deg(a₁),…,deg(a_n)}≤ B. The bound depends on d,r,s and the maximal degree of the coefficients of the f_j where j = 1,…,r. In particular, if F is finite, a common zero can be computed effectively.

Milestones

Received: 1 April 1996

Published: 1 September 1997