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Equivalence of F-algebras and cubic forms Manindra Agrawal and Nitin Saxena

Summary: Equivalence of F-algebras and cubic forms
Manindra Agrawal and Nitin Saxena
Department of Computer Science
IIT Kanpur, India
September 15, 2005
We study the isomorphism problem of two "natural" algebraic structures F-algebras and
cubic forms. We prove that the F-algebra isomorphism problem reduces in polynomial time
to the cubic forms equivalence problem. This answers a question asked in [AS05]. For finite
fields F with 3 |(#F - 1), this result implies that the two problems are infact equivalent.
This result also has the following interesting consequence:
Graph Isomorphism P
m F-algebra Isomorphism P
m Cubic Form Equivalence.
1 Introduction
For a field F, F-algebras are commutative rings of finite dimension over F. One of the fundamental
computational problems about F-algebras is to decide, given two such algebras, if they are
isomorphic. When F is an algebraically closed field, it follows from Hilbert's Nullstellensatz
[Bro87] that the problem can be decided in PSPACE. When F = R, the problem is in EEXP


Source: Agrawal, Manindra - Department of Computer Science and Engineering, Indian Institute of Technology Kanpur


Collections: Computer Technologies and Information Sciences