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Register a new userAverage-case complexity of the Euclidean algorithm with a fixed polynomial over a finite field
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We analyze the behavior of the Euclidean algorithm applied to pairs (g,f) of univariate nonconstant polynomials over a finite field F_q of q elements when the highest-degree polynomial g is fixed. Considering all the elements f of fixed degree, we establish asymptotically optimal bounds in terms of q for the number of elements f which are relatively prime with g and for the average degree of gcd(g,f). The accuracy of our estimates is confirmed by practical experiments. We also exhibit asymptotically optimal bounds for the average-case complexity of the Euclidean algorithm applied to pairs (g,f) as above.
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We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by equivalent classes of polynomials with prescribed coefficients. Simplified expressions are derived for some special cases. Our results extend some earlier results.
A 1993 result of Alon and Furedi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain Condition (D) on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further Generalized Alon-Furedi Theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon-Furedi. We then discuss the relationship between Alon-Furedi and results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon-Furedi Theorem and its generalization in terms of Reed--Muller type affine variety codes is shown which gives us the minimum Hamming distance of these codes. Then we apply the Alon-Furedi Theorem to quickly recover (and sometimes strengthen) old and new results in finite geometry, including the Jamison/Brouwer-Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.
We count the number of Coxeters friezes over a finite field. Our method uses geometric realizations of the spaces of friezes in a certain completion of the classical moduli space $mathcal{M}_{0,n}$ allowing repeated points in the configurations. Counting points in the completed moduli space over a finite field is related to the enumeration problem of counting partitions of cyclically ordered set of points into subsets containing no consecutive points. In Appendix we provide an elementary solution for this enumeration problem.
How many bits of information are revealed by a learning algorithm for a concept class of VC-dimension $d$? Previous works have shown that even for $d=1$ the amount of information may be unbounded (tend to $infty$ with the universe size). Can it be that all concepts in the class require leaking a large amount of information? We show that typically concepts do not require leakage. There exists a proper learning algorithm that reveals $O(d)$ bits of information for most concepts in the class. This result is a special case of a more general phenomenon we explore. If there is a low information learner when the algorithm {em knows} the underlying distribution on inputs, then there is a learner that reveals little information on an average concept {em without knowing} the distribution on inputs.
It is proved that with finitely many possible exceptions, each cyclotomic scheme over finite field is determined up to isomorphism by the tensor of 2-dimensional intersection numbers; for infinitely many schemes, this result cannot be improved. As a consequence, the Weisfeiler-Leman dimension of a Paley graph or tournament is at most 3 with possible exception of several small graphs.
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