Holzer’s theorem in k[t]

Abstract

Let a, b, c be nonzero polynomials in k[t] where k[t] is the ring of polynomials with coefficients in k. We prove that if $ax^2 \,{+}\, by^2 \,{+}\, cz^2 = 0$ has a nonzero solution in k[t], then there exist $x_0,y_0,z_0 \in k[t]$, not all zero, such that $ax_0^2 \,{+}\, by_0^2 \,{+}\, cz_0^2 = 0$ and $\deg x_0 \le \frac{1}{2}(\deg b \,{+}\, \deg c)$, $\deg y_0 \le \frac{1}{2}(\deg a \,{+}\, \deg c)$, and $\deg z_0 \le \frac{1}{2}(\deg a \,{+}\, \deg b)$. This is the polynomial analogue of Holzer’s theorem for $ax^2 \,{+}\, by^2 \,{+}\, cz^2 = 0$ when a, b, c are integers.