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}
}
• 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…
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References

SHOWING 1-10 OF 36 REFERENCES

• Computer Science, Mathematics
Acta Informatica
• 2005
This paper generalizes the well-known Sch6nhage-Strassen algorithm for multiplying large integers to an algorithm for dividing polynomials with coefficients from an arbitrary, not necessarily commutative, not always associative, algebra d, and obtains a method not requiring division that is valid for any algebra.
• Mathematics
ESA
• 2004
It is shown how to evaluate a bivariate polynomial p of maximum degree less than n, specified by its n 2 coefficients, simultaneously at n 2 given points using a total of $$\mathcal{O}(n^{2.667})$$ arithmetic operations.
• Computer Science
ArXiv
• 2021
A new Las Vegas algorithm is presented for the composition of two polynomials modulo a third one, over an arbitrary field, breaking through the $3/2$ barrier in the exponent for the first time.
• Computer Science
IEEE Transactions on Information Theory
• 2021
For codes over curves whose evaluation points lie on a grid-like structure, for example the Hermitian curve and norm-trace curves, it is shown that the encoding and unencoding algorithms have quasi-linear time complexity.
• Computer Science, Mathematics
J. Complex.
• 2020
This paper presents a new algorithm for reducing a multivariate polynomial with respect to an autoreduced tuple of other polynomials and shows that the execution time is essentially the same as the time needed to verify that the result is correct.
Necessary and sufficient conditions are given for projective plane curves to have cusps isomorphic to the origin of an affine plane curve xa + yb. Based on them, a family of curves Cab having only
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