The Ramanujan Journal

, Volume 48, Issue 2, pp 351–355

# Holzer’s theorem in k[t]

• José Luis Leal-Ruperto
• David B. Leep
Article

## Abstract

Let abc 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 abc are integers.

## Keywords

Legendre equation Rational polynomial ring Holzer’s theorem

11D 11G

## References

1. 1.
Holzer, L.: Minimal solutions of Diophantine equations. Can. J. Math. 2, 238–244 (1950)
2. 2.
Mordell, L.J.: On the magnitude of the integer solutions of the equation $$ax^2 +by^2 +cz^2 =0$$. J. Number Theory 1, 1–3 (1969)
3. 3.
Prestel, A.: On the size of zeros of quadratic forms over rational function fields. J. Reine Angew Math. 378, 101–112 (1987)