140,207 research outputs found
Counting real rational functions with all real critical values
We study the number of real rational degree n functions (considered up to
linear fractional transformations of the independent variable) with a given set
of 2n-2 distinct real critical values. We present a combinatorial reformulation
of this number and pose several related questions.Comment: 12 pages (AMSTEX), 3 picture
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=(p_1,...,p_n) are a subset of the
free variables in a Presburger formula, we can define a counting function g(p)
to be the number of solutions to the formula, for a given p. We show that every
counting function obtained in this way may be represented as, equivalently,
either a piecewise quasi-polynomial or a rational generating function. Finally,
we translate known computational complexity results into this setting and
discuss open directions.Comment: revised, including significant additions explaining computational
complexity results. To appear in Journal of Symbolic Logic. Extended abstract
in ICALP 2013. 17 page
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