Holzer’s theorem in k[t]

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 ax2+by2+cz2=0 has a nonzero solution in k[t], then there exist x, y, z∈ k[ t] , not all zero, such that ax02+by02+cz02=0 and degx0≤12(degb+degc), degy0≤12(dega+degc), and degz0≤12(dega+degb). This is the polynomial analogue of Holzer’s theorem for ax2+by2+cz2=0 when a, b, c are integers.