Linear isomorphisms of non-degenerate integral ternary cubic forms

Inhwan Lee; Byeong-Kweon Oh

doi:10.4134/BKMS.b150889

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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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

Linear isomorphisms of non-degenerate integral ternary cubic forms

Inhwan Lee and Byeong-Kweon Oh

Seoul National University, Seoul National University

Abstract

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Abstract

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