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 a x 2 + b y 2 + c z 2 = 0 has a nonzero solution in k [ t ], then there exist x 0 , y 0 , z 0 ∈ k [ t ] , not all zero, such that a x 0 2 + b y 0 2 + c z 0 2 = 0 and d...

Full description

Saved in:
Bibliographic Details
Published in:The Ramanujan journal 2019-02, Vol.48 (2), p.351-355
Main Authors: Leal-Ruperto, José Luis, Leep, David B.
Format: Article
Language:English
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Integers
Mathematics
Mathematics and Statistics
Number Theory
Polynomials
Rings (mathematics)
Theorems
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 a x 2 + b y 2 + c z 2 = 0 has a nonzero solution in k [ t ], then there exist x 0 , y 0 , z 0 ∈ k [ t ] , not all zero, such that a x 0 2 + b y 0 2 + c z 0 2 = 0 and deg x 0 ≤ 1 2 ( deg b + deg c ) , deg y 0 ≤ 1 2 ( deg a + deg c ) , and deg z 0 ≤ 1 2 ( deg a + deg b ) . This is the polynomial analogue of Holzer’s theorem for a x 2 + b y 2 + c z 2 = 0 when a ,  b ,  c are integers.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-017-9946-x