Linear isomorphisms of non-degenerate integral ternary cubic forms
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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.
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.
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