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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.
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A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real morsifications for real plane curve singularities having arbitrary real local branches and pairs of complex conjugate branches satisfying some conditions. This was known before only in the case of all local branches being real (ACampo, Gusein-Zade). We also discuss a relation between real morsifications and the topology of singularities, extending to arbitrary real morsifications the Balke-Kaenders theorem, which states that the ACampo--Gusein-Zade diagram associated to a morsification uniquely determines the topological type of a singularity.
We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a single n-variate m-nomial.
Let $F(x, y) in mathbb{C}[x,y]$ be a polynomial of degree $d$ and let $G(x,y) in mathbb{C}[x,y]$ be a polynomial with $t$ monomials. We want to estimate the maximal multiplicity of a solution of the system $F(x,y) = G(x,y) = 0$. Our main result is that the multiplicity of any isolated solution $(a,b) in mathbb{C}^2$ with nonzero coordinates is no greater than $frac{5}{2}d^2t^2$. We ask whether this intersection multiplicity can be polynomially bounded in the number of monomials of $F$ and $G$, and we briefly review some connections between sparse polynomials and algebraic complexity theory.
For each closed surface of genus $gge3$, we find a finite subcomplex of the separating curve complex that is rigid with respect to incidence-preserving maps.
We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $mathbb{P}^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.
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