37,271 research outputs found
Tropical Discriminants
Tropical geometry is used to develop a new approach to the theory of
discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky.
The tropical A-discriminant, which is the tropicalization of the dual variety
of the projective toric variety given by an integer matrix A, is shown to
coincide with the Minkowski sum of the row space of A and of the
tropicalization of the kernel of A. This leads to an explicit positive formula
for the extreme monomials of any A-discriminant, without any smoothness
assumption.Comment: Major revisions, including several improvements and the correction of
Section 5. To appear: Journal of the American Mathematical Societ
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
The Newton Polytope of the Implicit Equation
We apply tropical geometry to study the image of a map defined by Laurent
polynomials with generic coefficients. If this image is a hypersurface then our
approach gives a construction of its Newton polytope.Comment: 18 pages, 3 figure
New constructions for covering designs
A {\em covering design}, or {\em covering}, is a family of
-subsets, called blocks, chosen from a -set, such that each -subset is
contained in at least one of the blocks. The number of blocks is the covering's
{\em size}, and the minimum size of such a covering is denoted by .
This paper gives three new methods for constructing good coverings: a greedy
algorithm similar to Conway and Sloane's algorithm for lexicographic
codes~\cite{lex}, and two methods that synthesize new coverings from
preexisting ones. Using these new methods, together with results in the
literature, we build tables of upper bounds on for , , and .
A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins
The main result of this paper is a bijective proof showing that the
generating function for partitions with bounded differences between largest and
smallest part is a rational function. This result is similar to the closely
related case of partitions with fixed differences between largest and smallest
parts which has recently been studied through analytic methods by Andrews,
Beck, and Robbins. Our approach is geometric: We model partitions with bounded
differences as lattice points in an infinite union of polyhedral cones.
Surprisingly, this infinite union tiles a single simplicial cone. This
construction then leads to a bijection that can be interpreted on a purely
combinatorial level.Comment: 12 pages, 5 figure
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