On zeros of bounded degree of systems of homogeneous polynomial equations (Q1366640)

Let \(F\) be a finite or algebraically closed field and \(R=F[T_1,\dots, T_s]\), the polynomial ring in \(T_1,\dots, T_s\) over \(F\). By Tsen-Lang, any system of homogeneous polynomials \(f_1(X),\dots, f_r(X)\in R[X]\) of degree \(d\), where \(X=(X_1,\dots, X_n)\), has a non-trivial common zero in \(R^n\) provided the number of variables \(n\) is sufficiently large. In this note we give an effective bound \(B\) such that there exists a zero \(0\neq (a_1,\dots,a_n)\in R^n\) with \(\max\{\deg (a_1),\dots, \deg(a_n)\}\leq B\). The bound depends on \(d,r,s\) and the maximal degree of the coefficients of the \(f_j\), where \(j=1,\dots, r\). In particular, if \(F\) is finite, a common zero can be computed effectively.