Cyclotomy and permutation polynomials of large indices

We use cyclotomy to design new classes of permutation polynomials over finite fields. This allows us to generate many classes of permutation polynomials in an algorithmic way. Many of them are permutation polynomials of large indices

Algorithmic Polynomials

The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity. We develop a first-principles, classical approach to the polynomial approximation of Boolean functions. We use it to give the first constructive upper bounds on the approximate degree of several fundamental problems: - $O\bigl(n^{\frac{3}{4}-\frac{1}{4(2^{k}-1)}}\bigr)$ for the $k$-element distinctness problem; - $O(n^{1-\frac{1}{k+1}})$ for the $k$-subset sum problem; - $O(n^{1-\frac{1}{k+1}})$ for any $k$-DNF or $k$-CNF formula; - $O(n^{3/4})$ for the surjectivity problem. In all cases, we obtain explicit, closed-form approximating polynomials that are unrelated to the quantum arguments from previous work. Our first three results match the bounds from quantum query complexity. Our fourth result improves polynomially on the $\Theta(n)$ quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree $\Omega(n)$. In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity

Использование таблиц функций для быстрого вычисления многочленов

The table and algorithmic method of calculation of polynomials based on preliminary coefficient processing is offered. Possibility of acceleration of calculation of polynomials in comparison with realization of the well-known table and algorithmic methods is shown.Предложен таблично-алгоритмический метод вычисления многочленов, основанный на предварительной обработке коэффициентов. Показана возможность ускорения вычисления многочленов по сравнению с реализацией известных таблично-алгоритмических методов

Polar Varieties and Efficient Real Elimination

Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1, ..., f_p$. This paper is devoted to the algorithmic problem of finding {\em efficiently} a representative point for each connected component of $S_0$ . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of $S_0$. This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations $f_1, >..., f_p$ and in a suitably introduced, intrinsic geometric parameter, called the {\em degree} of the real interpretation of the given equation system $f_1, >..., f_p$.Comment: 32 page