Bull. Korean Math. Soc. 2016; 53(6): 1697-1705
Online first article September 22, 2016 Printed November 30, 2016
https://doi.org/10.4134/BKMS.b150889
Copyright © The Korean Mathematical Society.
Inhwan Lee and Byeong-Kweon Oh
Seoul National University, Seoul National University
In this article, we consider the problem on finding non-degen\-erate $n$-ary $m$-ic forms having an $n \times n$ matrix $A$ as a linear isomorphism. We show that it is equivalent to solve a linear diophantine equation. In particular, we find all integral ternary cubic forms having $A$ as a linear isomorphism, for any $A \in GL_3(\z)$. We also give a family of non-degenerate cubic forms $F$ such that $F(\mathbf x)=N$ always has infinitely many integer solutions if exists.
Keywords: linear isomorphisms, $n$-ary $m$-ic forms
MSC numbers: Primary 11E76, 15A63, 11D25
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