 
Summary: Equivalence of Falgebras and cubic forms
Manindra Agrawal and Nitin Saxena
Department of Computer Science
IIT Kanpur, India
{manindra,nitinsa}@cse.iitk.ac.in
September 15, 2005
Abstract
We study the isomorphism problem of two "natural" algebraic structures Falgebras and
cubic forms. We prove that the Falgebra 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 Falgebra Isomorphism P
m Cubic Form Equivalence.
1 Introduction
For a field F, Falgebras are commutative rings of finite dimension over F. One of the fundamental
computational problems about Falgebras 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
