356 research outputs found
Liminal reciprocity and factorization statistics
Let denote the number of monic irreducible polynomials in
of degree . We show that for a fixed
degree , the sequence converges -adically to an explicitly
determined rational function . Furthermore we show that the
limit is related to the classic necklace polynomial
by an involutive functional equation, leading to a phenomenon we
call liminal reciprocity. The limiting first moments of factorization
statistics for squarefree polynomials are expressed in terms of a family of
symmetric group representations as a consequence of liminal reciprocity.Comment: 22 pages. To appear in Algebraic Combinatoric
The Galois theory of the lemniscate
This article studies the Galois groups that arise from division points of the
lemniscate. We compute these Galois groups two ways: first, by class field
theory, and second, by proving the irreducibility of lemnatomic polynomials,
which are analogs of cyclotomic polynomials. We also discuss Abel's theorem on
the lemniscate and explain how lemnatomic polynomials relate to Chebyshev
polynomials.Comment: The revised version adds four references and some historical remarks.
We also note that a special case of Theorem 4.1 appears in Lemmermeyer's
Reciprocity Law
Pre-images of quadratic dynamical systems
For a quadratic endomorphism of the affine line defined over the rationals we
consider the problem of bounding the number of rational points that eventually
land at a given constant after iteration, called pre-images of the constant. In
the article "Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems,"
it was shown that the number of rational pre-images is bounded as one varies
the morphism in a certain one-dimensional family. Explicit values of the
constant for pre-images of zero and -1 defined over the rational numbers were
addressed in subsequent articles. This article addresses an explicit bound for
any algebraic image constant and provides insight into the geometry of the
"pre-image surfaces."Comment: to appear in Involve; 16page
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