1,296 research outputs found
LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future
10 years ago or so Bill Helton introduced me to some mathematical problems
arising from semidefinite programming. This paper is a partial account of what
was and what is happening with one of these problems, including many open
questions and some new results
Livsic-type Determinantal Representations and Hyperbolicity
Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic
real projective hypersurfaces, and their determinantal representations, play a
key role in the emerging field of convex algebraic geometry. In this paper we
consider a natural notion of hyperbolicity for a real subvariety of an arbitrary codimension with respect to a real -dimensional linear subspace and study its basic
properties. We also consider a special kind of determinantal representations
that we call Livsic-type and a nice subclass of these that we call \vr{}. Much
like in the case of hypersurfaces (), the existence of a definite
Hermitian \vr{} Livsic-type determinantal representation implies hyperbolicity.
We show that every curve admits a \vr{} Livsic-type determinantal
representation. Our basic tools are Cauchy kernels for line bundles and the
notion of the Bezoutian for two meromorphic functions on a compact Riemann
surface that we introduce. We then proceed to show that every real curve in
hyperbolic with respect to some real -dimensional linear
subspace admits a definite Hermitian, or even real symmetric, \vr{} Livsic-type
determinantal representation
Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support)
Consider rectangular matrices over a commutative ring R. Assume the ideal of
maximal minors factorizes, I_m(A)=J_1*J_2. When is A left-right equivalent to a
block-diagonal matrix? (When does the module/sheaf Coker(A) decompose as the
corresponding direct sum?) If R is not a principal ideal ring (or a close
relative of a PIR) one needs additional assumptions on A. No necessary and
sufficient criterion for such block-diagonal reduction is known.
In this part we establish the following:
* The persistence of (in)decomposability under the change of rings. For
example, the passage to Noetherian/local/complete rings, the decomposability of
A over a graded ring R vs the decomposability of Coker(A) locally at the points
of Proj(R), the restriction to a subscheme in Spec(R).
* The necessary and sufficient condition for decomposability of square
matrices in the case: det(A)=f_1*f_2 is not a zero divisor and f_1,f_2 are
co-prime.
As an immediate application we give criteria of simultaneous (block-)diagonal
reduction for tuples of matrices over a field, i.e. linear determinantal
representations
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