We consider the problem of solving systems of multivariate polynomial equations of degree k over a finite field. For every integer k ≥ 2 and finite field F q where q = p d for a prime p, we give, to the best of our knowledge, the first algorithms that achieve an exponential speedup over the brute force O(q n ) time algorithm in the worst case. We present two algorithms, a randomized algorithm with running time q n+o (n) • q −n/O(k) time if q ≤ 2 4ekd , and q n+o(n) • ( log q dek ) −dn otherwise, where e = 2.718 . . . is Napier's constant, and a deterministic algorithm for counting solutions with running time q n+o (n) • q −n/O(kq 6/7d ) . For the important special case of quadratic equations in F 2 , our randomized algorithm has running time O(2 0.8765n ).For systems over F 2 we also consider the case where the input polynomials do not have bounded degree, but instead can be efficiently represented as a ΣΠΣ circuit, i.e., a sum of products of sums of variables. For this case we present a deterministic algorithm running in time 2 n−δ n for δ = 1/O(log(s/n)) for instances with s product gates in total and n variables.Our algorithms adapt several techniques recently developed via the polynomial method from circuit complexity. The algorithm for systems of ΣΠΣ polynomials also introduces a new degree reduction method that takes an instance of the problem and outputs a subexponential-sized set of instances, in such a way that feasibility is preserved and every polynomial among the output instances has degree O(log(s/n)).
We say that an algorithm robustly decides a constraint satisfaction problem Π if it distinguishes at-least-(1 − )-satisfiable instances from less-than-(1 − r( ))-satisfiable instances for some function r( ) with r( ) → 0 as → 0. In this paper we show that the canonical linear programming relaxation robustly decides Π if and only if Π has "width 1" (in the sense of Feder and Vardi).
Epidemiological and animal model studies have suggested that high intake of heme, present in red meat, is associated with an increased risk of colon cancer. However, the mechanisms underlying this association are not clear. This study aimed to investigate whether heme induces DNA damage and cell proliferation of colonic epithelial cells via hydrogen peroxide produced by heme oxygenase (HO). We examined the effects of zinc protoporphyrin (ZnPP; a HO inhibitor) and catalase on DNA damage, cell proliferation, and IL-8 production induced by the addition of hemin (1-10 microM) to human colonic epithelial Caco-2 cells. DNA damage was determined with a comet assay, and cell proliferation was evaluated with 5-bromo-2'-deoxyuridine incorporation assay. Both ZnPP and exogenous catalase inhibited the hemin-induced DNA damage and cell hyperproliferation dose-dependently. IL-8 messenger RNA expression and IL-8 production in the epithelial cells increased following the hemin treatment, but the production was inhibited by ZnPP and catalase. These results indicate that hemin has genotoxic and hyperproliferative effects on Caco-2 cells by HO and hydrogen peroxide. The mechanism might explain why a high intake of heme is associated with increased risk of colon cancer.
We present a moderately exponential time algorithm for the satisfiability of Boolean formulas over the full binary basis. For formulas of size at most cn, our algorithm runs in time 2 (1−µc)n for some constant µ c > 0. As a byproduct of the running time analysis of our algorithm, we obtain strong average-case hardness of affine extractors for linear-sized formulas over the full binary basis.
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