# 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} }

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…

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