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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.
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참고문헌 (10)
참고문헌 신청
M. Drmota / 1995 / Relations between polynomial roots / Acta Arith 71(1):65~77
A. Elsenhans / 2009 / New sum of three cubes / Math. Comp 78(266):1227~1230
R. A. Horn / 1999 / The Jordan canonical forms of complex orthogonal and skew-symmetric matrices / Linear Algebra Appl 302/303:411~421
C. Jordan / 1880 / Memore sur l’equivalence des formes / J. Ec. Pol. XLVIII :112~150
M.-H. Kim / 2000 / Generation of isometries of certain Z-lattices by symmetries / J. Number Theory 83(1):76~90
A. Elsenhans / 2009 / New sum of three cubes / Math. Comp 78(266):1227~1230
R. A. Horn / 1999 / The Jordan canonical forms of complex orthogonal and skew-symmetric matrices / Linear Algebra Appl 302/303:411~421
C. Jordan / 1880 / Memore sur l’equivalence des formes / J. Ec. Pol. XLVIII :112~150
M.-H. Kim / 2000 / Generation of isometries of certain Z-lattices by symmetries / J. Number Theory 83(1):76~90
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