Abstract
We study the isomorphism problem of two “natural” algebraic structures – \(\mathbb{F}\)-algebras and cubic forms. We prove that the \(\mathbb{F}\)-algebra isomorphism problem reduces in polynomial time to the cubic forms equivalence problem. This answers a question asked in [AS05]. For finite fields of the form \(3 \Lambda(\#\mathbb{F} - 1)\), this result implies that the two problems are infact equivalent. This result also has the following interesting consequence:
Graph Isomorphism \({\leq}^P_m\) \(\mathbb{F}\)-algebra Isomorphism \({\leq}^P_m\) Cubic Form Equivalence.
This work was done while the authors were visiting National University of Singapore. The second author was also supported by research funding from Infosys Technologies Limited, Bangalore.
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Agrawal, M., Saxena, N. (2006). Equivalence of \(\mathbb{F}\)-Algebras and Cubic Forms. In: Durand, B., Thomas, W. (eds) STACS 2006. STACS 2006. Lecture Notes in Computer Science, vol 3884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11672142_8
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DOI: https://doi.org/10.1007/11672142_8
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