• DOI:10.1145/2930889.2930936
• Corpus ID: 11916304

Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

@article{Neiger2016FastCO,
  title={Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations},
  author={Vincent Neiger},
  journal={Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation},
  year={2016}
}

• Vincent Neiger
• Published 1 February 2016
• Mathematics, Computer Science
• Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

We give a Las Vegas algorithm which computes the shifted Popov form of an m x m nonsingular polynomial matrix of degree d in expected ~O(mω d) field operations, where ω is the exponent of matrix multiplication and ~O(·) indicates that logarithmic factors are omitted. This is the first algorithm in ~O(mω d) for shifted row reduction with arbitrary shifts. Using partial linearization, we reduce the problem to the case d ≤ ⌈ σ/m ⌉ where σ is the generic determinant bound, with σ / m bounded from above by both the average row degree and the average column degree of the matrix. The cost above becomes ~O(mω ⌈ σ/m ⌉), improving upon the cost of the fastest previously known algorithm for row reduction, which is deterministic. Our algorithm first builds a system of modular equations whose solution set is the row space of the input matrix, and then finds the basis in shifted Popov form of this set. We give a deterministic algorithm for this second step supporting arbitrary moduli in ~O(mω-1 σ) field operations, where m is the number of unknowns and σ is the sum of the degrees of the moduli. This extends previous results with the same cost bound in the specific cases of order basis computation and M-Pade approximation, in which the moduli are products of known linear factors.