29,732 research outputs found
The monic integer transfinite diameter
We study the problem of finding nonconstant monic integer polynomials,
normalized by their degree, with small supremum on an interval I. The monic
integer transfinite diameter t_M(I) is defined as the infimum of all such
supremums. We show that if I has length 1 then t_M(I) = 1/2.
We make three general conjectures relating to the value of t_M(I) for
intervals I of length less that 4. We also conjecture a value for t_M([0, b])
where 0 < b < 1. We give some partial results, as well as computational
evidence, to support these conjectures.
We define two functions that measure properties of the lengths of intervals I
with t_M(I) on either side of t. Upper and lower bounds are given for these
functions.
We also consider the problem of determining t_M(I) when I is a Farey
interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning
this value is true for an infinite family of Farey intervals.Comment: 32 pages, 5 figure
Parallel Integer Polynomial Multiplication
We propose a new algorithm for multiplying dense polynomials with integer
coefficients in a parallel fashion, targeting multi-core processor
architectures. Complexity estimates and experimental comparisons demonstrate
the advantages of this new approach
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
We consider the problem of finding the number of matrices over a finite field
with a certain rank and with support that avoids a subset of the entries. These
matrices are a q-analogue of permutations with restricted positions (i.e., rook
placements). For general sets of entries these numbers of matrices are not
polynomials in q (Stembridge 98); however, when the set of entries is a Young
diagram, the numbers, up to a power of q-1, are polynomials with nonnegative
coefficients (Haglund 98).
In this paper, we give a number of conditions under which these numbers are
polynomials in q, or even polynomials with nonnegative integer coefficients. We
extend Haglund's result to complements of skew Young diagrams, and we apply
this result to the case when the set of entries is the Rothe diagram of a
permutation. In particular, we give a necessary and sufficient condition on the
permutation for its Rothe diagram to be the complement of a skew Young diagram
up to rearrangement of rows and columns. We end by giving conjectures
connecting invertible matrices whose support avoids a Rothe diagram and
Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl
Small-span Hermitian matrices over quadratic integer rings
Building on the classification of all characteristic polynomials of integer
symmetric matrices having small span (span less than 4), we obtain a
classification of small-span polynomials that are the characteristic polynomial
of a Hermitian matrix over some quadratic integer ring. Taking quadratic
integer rings as our base, we obtain as characteristic polynomials some
low-degree small-span polynomials that are not the characteristic (or minimal)
polynomial of any integer symmetric matrix.Comment: 16 page
Dense Fewnomials
We derive new bounds of fewnomial type for the number of real solutions to
systems of polynomials that have structure intermediate between fewnomials and
generic (dense) polynomials. This uses a modified version of Gale duality for
polynomial systems. We also use stratified Morse theory to bound the total
Betti number of a hypersurface defined by such a dense fewnomial. These bounds
contain and generalize previous bounds for ordinary fewnomials obtained by
Bates, Bertrand, Bihan, and Sottile.Comment: 20 page
- …