1,592 research outputs found

    Near-optimal Bootstrapping of Hitting Sets for Algebraic Models

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    The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel [Ore22,DL78,Zip79,Sch80] states that any nonzero polynomial f(x1,,xn)f(x_1,\ldots, x_n) of degree at most ss will evaluate to a nonzero value at some point on a grid SnFnS^n \subseteq \mathbb{F}^n with S>s|S| > s. Thus, there is an explicit hitting set for all nn-variate degree ss, size ss algebraic circuits of size (s+1)n(s+1)^n. In this paper, we prove the following results: - Let ϵ>0\epsilon > 0 be a constant. For a sufficiently large constant nn and all s>ns > n, if we have an explicit hitting set of size (s+1)nϵ(s+1)^{n-\epsilon} for the class of nn-variate degree ss polynomials that are computable by algebraic circuits of size ss, then for all ss, we have an explicit hitting set of size sexpexp(O(logs))s^{\exp \circ \exp (O(\log^\ast s))} for ss-variate circuits of degree ss and size ss. That is, if we can obtain a barely non-trivial exponent compared to the trivial (s+1)n(s+1)^{n} sized hitting set even for constant variate circuits, we can get an almost complete derandomization of PIT. - The above result holds when "circuits" are replaced by "formulas" or "algebraic branching programs". This extends a recent surprising result of Agrawal, Ghosh and Saxena [AGS18] who proved the same conclusion for the class of algebraic circuits, if the hypothesis provided a hitting set of size at most (sn0.5δ)(s^{n^{0.5 - \delta}}) (where δ>0\delta>0 is any constant). Hence, our work significantly weakens the hypothesis of Agrawal, Ghosh and Saxena to only require a slightly non-trivial saving over the trivial hitting set, and also presents the first such result for algebraic branching programs and formulas.Comment: The main result has been strengthened significantly, compared to the older version of the paper. Additionally, the stronger theorem now holds even for subclasses of algebraic circuits, such as algebraic formulas and algebraic branching program

    Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra

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    We present algorithms to factorize weighted homogeneous elements in the first polynomial Weyl algebra and qq-Weyl algebra, which are both viewed as a Z\mathbb{Z}-graded rings. We show, that factorization of homogeneous polynomials can be almost completely reduced to commutative univariate factorization over the same base field with some additional uncomplicated combinatorial steps. This allows to deduce the complexity of our algorithms in detail. Furthermore, we will show for homogeneous polynomials that irreducibility in the polynomial first Weyl algebra also implies irreducibility in the rational one, which is of interest for practical reasons. We report on our implementation in the computer algebra system \textsc{Singular}. It outperforms for homogeneous polynomials currently available implementations dealing with factorization in the first Weyl algebra both in speed and elegancy of the results.Comment: 26 pages, Singular implementation, 2 algorithms, 1 figure, 2 table

    Survey on counting special types of polynomials

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    Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size. Furthermore, a univariate polynomial f is decomposable if f = g o h for some nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. The crux of the matter is to count the number of collisions, where essentially different (g, h) yield the same f. We present a classification of all collisions at degree n = p^2 which yields an exact count of those decomposable polynomials.Comment: to appear in Jaime Gutierrez, Josef Schicho & Martin Weimann (editors), Computer Algebra and Polynomials, Lecture Notes in Computer Scienc
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