78 research outputs found
Two results on the size of spectrahedral descriptions
A spectrahedron is a set defined by a linear matrix inequality. Given a
spectrahedron we are interested in the question of the smallest possible size
of the matrices in the description by linear matrix inequalities. We show
that for the -dimensional unit ball is at least . If
, then we actually have . The same holds true for any compact
convex set in defined by a quadratic polynomial. Furthermore, we
show that for a convex region in whose algebraic boundary is
smooth and defined by a cubic polynomial we have that is at least five.
More precisely, we show that if are real symmetric matrices such that
is a cubic polynomial, the surface in complex
projective three-space with affine equation is singular.Comment: 10 pages, 2 figures, minor mistakes correcte
A Note on the Hyperbolicity Cone of the Specialized V\'amos Polynomial
The specialized V\'amos polynomial is a hyperbolic polynomial of degree four
in four variables with the property that none of its powers admits a definite
determinantal representation. We will use a heuristical method to prove that
its hyperbolicity cone is a spectrahedron.Comment: Notable easier arguments and minor correction
The separating semigroup of a real curve
We introduce the separating semigroup of a real algebraic curve of dividing
type. The elements of this semigroup record the possible degrees of the
covering maps obtained by restricting separating morphisms to the real part of
the curve. We also introduce the hyperbolic semigroup which consists of
elements of the separating semigroup arising from morphisms which are
compositions of a linear projection with an embedding of the curve to some
projective space.
We completely determine both semigroups in the case of maximal curves. We
also prove that any embedding of a real curve to projective space of
sufficiently high degree is hyperbolic. Using these semigroups we show that the
hyperbolicity locus of an embedded curve is in general not connected.Comment: 14 pages, 4 figures, published version, comments welcome
Interlacing Ehrhart Polynomials of Reflexive Polytopes
It was observed by Bump et al. that Ehrhart polynomials in a special family
exhibit properties similar to the Riemann {\zeta} function. The construction
was generalized by Matsui et al. to a larger family of reflexive polytopes
coming from graphs. We prove several conjectures confirming when such
polynomials have zeros on a certain line in the complex plane. Our main new
method is to prove a stronger property called interlacing
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