Multiplicative equations over commuting matrices

Conference · Tue Dec 31 04:00:00 EST 1996

OSTI ID:416835

Babai, L ^[1]; Beals, R ^[2]; Cai, Jin-Yi ^[3]

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1. Univ. of Chicago, IL (United States)
2. Rutgers Univ., Piscataway, NJ (United States)
3. SUNY, Buffalo, NY (United States); and others

We consider the solvability of the equation and generalizations, where the A{sub i} and B are given commuting matrices over an algebraic number field F. In the semigroup membership problem, the variables x{sub i} are constrained to be nonnegative integers. While this problem is NP-complete for variable k, we give a polynomial time algorithm if k is fixed. In the group membership problem, the matrices are assumed to be invertible, and the variables x{sub i} may take on negative values. In this case we give a polynomial time algorithm for variable k and give an explicit description of the set of all solutions (as an affine lattice). The special case of 1 x 1 matrices was recently solved by Guoqiang Ge; we heavily rely on his results.

• OSTI ID: 416835
• Report Number(s): CONF-960121--
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
ALGEBRA
ALGORITHMS
COMMUTATION RELATIONS
EQUATIONS
MATRICES