• DOI:10.1145/3373207.3404032
• Corpus ID: 214693152

Generic bivariate multi-point evaluation, interpolation and modular composition with precomputation

@article{Neiger2020GenericBM,
  title={Generic bivariate multi-point evaluation, interpolation and modular composition with precomputation},
  author={Vincent Neiger and Johan Sebastian Rosenkilde and Grigory Solomatov},
  journal={Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation},
  year={2020}
}

• Vincent Neiger, J. Rosenkilde, Grigory Solomatov
• Published 27 March 2020
• Mathematics, Computer Science
• Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation

Suppose K is a large enough field and P ⊂ K2 is a fixed, generic set of points which is available for precomputation. We introduce a technique called reshaping which allows us to design quasi-linear algorithms for both: computing the evaluations of an input polynomial f ∈ K [x, y] at all points of P and computing an interpolant f ∈ K[x, y] which takes prescribed values on P and satisfies an input y-degree bound. Our genericity assumption is explicit and we prove that it holds for most point sets over a large enough field. If P violates the assumption, our algorithms still work and the performance degrades smoothly according to a distance from being generic. To show that the reshaping technique may have an impact on other related problems, we apply it to modular composition: suppose generic polynomials M ∈ K[x] and A ∈ K[x] are available for precomputation, then given an input f ∈ K[x, y] we show how to compute f (x, A(x)) rem M(x) in quasi-linear time.