Fast algorithms for the characteristic polynomial
Theoretical Computer Science
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Fast rectangular matrix multiplication and applications
Journal of Complexity
Lower bounds for dynamic algebraic problems
Information and Computation
Dynamic Transitive Closure via Dynamic Matrix Inverse (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Deterministic network coding by matrix completion
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Algebraic Complexity Theory
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Fast matrix rank algorithms and applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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We consider maintaining information about the rank of a matrix under changes of the entries. For n×n matrices, we show an upper bound of O(n1.575) arithmetic operations and a lower bound of Ω(n) arithmetic operations per change. The upper bound is valid when changing up to O(n0.575) entries in a single column of the matrix. Both bounds appear to be the first non-trivial bounds for the problem. The upper bound is valid for arbitrary fields, whereas the lower bound is valid for algebraically closed fields. The upper bound uses fast rectangular matrix multiplication, and the lower bound involves further development of an earlier technique for proving lower bounds for dynamic computation of rational functions.