Rational parking functions and LLT polynomials

Computer Algebra of Polynomials and Rational Functions

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Faber Polynomials for Rational Mapping Functions

VC Dimension and Learnability of Sparse Polynomials and Rational Functions

We prove upper and lower bounds on the VC dimension of sparse univariate polynomi-als over reals, and apply these results to prove uniform learnability of sparse polynomials and rational functions. As another application we solve an open problem of Vapnik ((Vap-nik 82]) on uniform approximation of the general regression functions, a central problem of computational statistics (cf. Vapnik 82]), ...

Small Dynamical Heights for Quadratic Polynomials and Rational Functions

Let φ ∈ Q(z) be a polynomial or rational function of degree 2. A special case of Morton and Silverman’s Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of φ is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height ĥφ(x) of a non-preperiodic rational point x is bounded below by a uniform multiple o...

Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

Presburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p = (p1, . . . , pn) are a subset of the free...