# 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} }

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

### Ideal Bases and Primary Decomposition: Case of Two Variables

- MathematicsJ. Symb. Comput.
- 1985

### On fast multiplication of polynomials over arbitrary algebras

- Computer Science, MathematicsActa 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.

### Fast Multipoint Evaluation of Bivariate Polynomials

- MathematicsESA
- 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.

### Faster Modular Composition

- Computer ScienceArXiv
- 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.

### Fast Encoding of AG Codes Over Cab Curves

- Computer ScienceIEEE 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.

### On the Complexity of Multivariate Polynomial Division

- Computer Science, Mathematics
- 2015

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.

### Algebraic geometric codes on certain plane curves

- Mathematics
- 1993

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…

### Designs And Their Codes

- Computer Science, Art
- 2016

The designs and their codes is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can get it instantly.