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Session 30. Real Algebraic Geometry, applications and related topics
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Affine Plane Curves and Determinantal Representations with Maximal Signature |
Christoph Hanselka, University of Konstanz, Germany
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A symmetric (selfadjoint) determinantal representation of a real
homogeneous polynomial is a symmetric (selfadjoint) linear matrix
polynomial, whose determinant is the given polynomial. In the case
of ternary forms, these have been studied to a great extend for
example by Vinnikov in 1993 and particularly the relation between
the topology of the real zero locus and the maximal possible
signature of a representation at a given point. We will consider a
more elementary approach to an affine analog of this problem from
which we obtain as a special case also Vinnikov's result on the
existence of definite representations of hyperbolic polynomials.
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