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Register a new userOn the connectivity of the hyperbolicity region of irreducible polynomials
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Authors
Mario Kummer
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We give an elementary proof for the fact that an irreducible hyperbolic polynomial has only one pair of hyperbolicity cones.
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The aim of the paper is to produce new families of irreducible polynomials, generalizing previous results in the area. One example of our general result is that for a near-separated polynomial, i.e., polynomials of the form $F(x,y)=f_1(x)f_2(y)-f_2(x)f_1(y)$, then $F(x,y)+r$ is always irreducible for any constant $r$ different from zero. We also provide the biggest known family of HIP polynomials in several variables. These are polynomials $p(x_1,ldots,x_n) in K[x_1,ldots,x_n]$ over a zero characteristic field $K$ such that $p(h_1(x_1),ldots,h_n(x_n))$ is irreducible over $K$ for every $n$-tuple $h_1(x_1),ldots,h_n(x_n)$ of non constant one variable polynomials over $K$. The results can also be applied to fields of positive characteristic, with some modifications.
A bad point of a positive semidefinite real polynomial f is a point at which a pole appears in all expressions of f as a sum of squares of rational functions. We show that quartic polynomials in three variables never have bad points. We give examples of positive semidefinite polynomials with a bad point at the origin, that are nevertheless sums of squares of formal power series, answering a question of Brumfiel. We also give an example of a positive semidefinite polynomial in three variables with a complex bad point that is not real, answering a question of Scheiderer.
We study the hyperbolicity of singular quotients of bounded symmetric domains. We give effective criteria for such quotients to satisfy Green-Griffiths-Langs conjectures in both analytic and algebraic settings. As an application, we show that Hilbert modular varieties, except for a few possible exceptions, satisfy all expected conjectures.
We describe new irreducible components of the moduli space of rank $2$ semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes $(c_1,c_2,c_3)=(-1,2n,0)$ and $(c_1,c_2,c_3)=(0,n,0)$ always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, and compute the exact number of irreducible components of the moduli spaces of rank 2 semistable torsion free sheaves with Chern classes $(c_1,c_2,c_3)=(-1,2,m)$ for all possible values for $m$; all components turn out to be rational. Furthermore, we also prove that these moduli spaces are connected, showing that some of sheaves here considered are smoothable.
We compute the compactly supported Euler characteristic of the space of degree $d$ irreducible polynomials in $n$ variables with real coefficients and show that the values are given by the digits in the so-called balanced binary expansion of the number of variables $n$.
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