Given a proper cone $K \subseteq \mathbb{R}^n$, a multivariate polynomial $f \in \mathbb{C}[z] = \mathbb{C}[z_1, \ldots, z_n]$ is called $K$-stable if it does not have a root whose vector of the imaginary parts is contained in the interior of $K$. If $K$ is the non-negative orthant, then $K$-stability specializes to the usual notion of stability of polynomials.
We study conditions and certificates for the $K$-stability of a given polynomial $f$, especially for the case of determinantal polynomials as well as for quadratic polynomials. A particular focus is on psd-stability. For cones $K$ with a spectrahedral representation, we construct a semidefinite feasibility problem, which, in the case of feasibility, certifies $K$-stability of $f$. This reduction to a semidefinite problem builds upon techniques from the connection of containment of spectrahedra and positive maps.
In the case of psd-stability, if the criterion is satisfied, we can explicitly construct a determinantal representation of the given polynomial. We also show that under certain conditions, for a $K$-stable polynomial $f$, the criterion is at least fulfilled for some scaled version of $K$.
Aiming at a generalization of a classical theorem of Moebius, we study maps that take line intervals to plane curves, and also maps that take line intervals to conics from certain linear systems.
Given a holomorphic conic bundle without sections, we show that finite groups acting by its fiberwise bimeromorphic transformations are bounded. This provides an analog of a similar result obtained by T.Bandman and Yu.Zarhin for quasi-projective conic bundles.
We introduce and study the notion of conic stability of multivariate complex polynomials in $\mathbb{C}[z_1,\ldots, z_n]$, which naturally generalizes the stability of multivariate polynomials. In particular, we generalize Borcea's and Brändén's multivariate version of the Hermite-Kakeya-Obreschkoff Theorem to the conic stability and provide a characterization in terms of a directional Wronskian. And we generalize a major criterion for stability of determinantal polynomials to stability with respect to the positive semidefinite cone.
Given two nonsingular real algebraic varieties V and W, we consider the problem of deciding whether a smooth map f: V -> W can be approximated by regular maps in the space of smooth maps from V to W. Our main result is a complete solution to this problem in case W is the usual 2-dimensional sphere and V is a real algebraic surface of negative Kodaira dimension.
We complete Mori's program for Kontsevich's moduli space of degree 2 stable maps to Grassmannian of lines. We describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties, and Kirwan's partial desingularization.
Birational Yang-Baxter maps (`set-theoretical solutions of the Yang-Baxter equation') are considered. A birational map $(x,y)\mapsto(u,v)$ is called quadrirational, if its graph is also a graph of a birational map $(x,v)\mapsto(u,y)$. We obtain a classification of quadrirational maps on $\CP^1\times\CP^1$, and show that all of them satisfy the Yang-Baxter equation. These maps possess a nice geometric interpretation in terms of linear pencil of conics, the Yang-Baxter property being interpreted as a new incidence theorem of the projective geometry of conics.
The decomposition group of an irreducible plane curve $X\subset\mathbb P^2$ is the subgroup $\mathrm{Dec}(X)\subset\mathrm{Bir}(\mathbb P^2)$ of birational maps which restrict to a birational map of $X$. We show that $\mathrm{Dec}(X)$ is generated by its elements of degree $\leq2$ when $X$ is either a conic or rational cubic curve.
We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if $X$ is birational to a conic bundle.
We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining polynomial. Finally, we characterize those degree--two tropical polynomials which are reducible and factorize them. We show that there exist irreducible degree--two tropical polynomials giving rise to pairs of tropical lines.
On the Alexander polynomials of conic-line arrangements
Université Côte d’Azur, CNRS, LJAD（蔚蓝海岸大学）；Simion Stoilow Institute of Mathematics（西蒙·斯托伊洛瓦数学研究所）；University of the National Education Commission Krakow（克拉科夫国民教育委员会大学）；Ovidius University（奥维迪乌斯大学）
Let $k$ be a field with char $k \not= 2$, $X$ be an affine surface defined by the equation $z^2=P(x)y^2+Q(x)$ where $P(x), Q(x) \in k[x]$ are separable polynomials. We will investigate the rationality problem of $X$ in terms of the polynomials $P(x)$ and $Q(x)$. The necessary and sufficient condition is $s \leq 3$ with minor exceptions, where $s=s_1+s_2+s_3+s_4$, $s_1$ (resp. $s_2$, resp. $s_3$) being the number of $c \in \overline{k}$ such that $P(c)=0$ and $Q(c) \not\in k(c)^2$ (resp. $Q(c)=0$ and $P(c) \not\in k(c)^2$, resp. $P(c)=Q(c)=0$ and $-\frac{Q}{P}(c) \not\in k(c)^2$). $s_4=0$ or $1…
Mutations of the cluster variables generating the cluster algebra of type $A^{(2)}_2$ reduce to a two-dimensional discrete integrable system given by a quartic birational map. The invariant curve of the map is a singular quartic curve, and its resolution of the singularity induces a discrete integrable system on a conic governed by a cubic birational map conjugate to the cluster mutations of type $A^{(2)}_2$. Moreover, it is shown that the conic is also the invariant curve of the quadratic birational map arising from the cluster mutations of type $A^{(1)}_1$ and the two birational maps on the …
In this paper, we compute the contributions from double cover maps to genus 0 degree 2 Gromov-Witten invariants of general type projective hypersurfaces. Our results correspond to a generalization of Aspinwall-Morrison formula to general type hypersurfaces in some special cases.
Computation of parallel lines (envelopes) to parabolas, ellipses, and hyperbolas is of importance in structure engineering and theory of mechanisms. Homogeneous polynomials that implicitly define parallel lines for the given offset to a conic are found by computing Groebner bases for an elimination ideal of a suitably defined affine variety. Singularity of the lines is discussed and their singular points are explicitly found as functions of the offset and the parameters of the conic. Critical values of the offset are linked to the maximum curvature of each conic. Application to a finite elemen…
Given a multivariate complex polynomial ${p\in\mathbb{C}[z_1,\ldots,z_n]}$, the imaginary projection $\mathcal{I}(p)$ of $p$ is defined as the projection of the variety $\mathcal{V}(p)$ onto its imaginary part. We focus on studying the imaginary projection of complex polynomials and we state explicit results for certain families of them with arbitrarily large degree or dimension. Then, we restrict to complex conic sections and give a full characterization of their imaginary projections, which generalizes a classification for the case of real conics. That is, given a bivariate complex polynomia…
We present a list of all polynomial dominanting maps of the complex plane with branched value curve isomorphic to the complex line, up to polynomial automorphisms.
We classify all quadratic homogeneous polynomial maps $H$ and Keller maps of the form $x + H$, for which $rk J H = 3$, over a field $K$ of arbitrary characteristic. In particular, we show that such a Keller map (up to a square part if $char K=2$) is a tame automorphism.
By using Oprea's Bialynicki-Birula decomposition for the stack of genus zero stable maps to flag manifolds. We calculate the Poincaré polynomial of the moduli space in degree one and degree two.
In this article we analyze the global diffeomorphism property of polynomial maps $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials $\|F\|_2^2$. This allows us to identify a class of polynomial maps $F$ for which their global diffeomorphism property on $\mathbb{R}^n$ is equivalent to their Jacobian determinant $\text{det }JF$ vanishing nowhere on $\mathbb{R}^n$. In other words, we identify a class of polynomial maps for which the Real Jacobian Conjecture, which was proven to be false in general,…
In the paper, we first classify all polynomial maps of the form $H=(u(x,y),v(x,y,z), h(x,y))$ in the case that $JH$ is nilpotent and $(°_yu,°_yh)\leq 3$, $H(0)=0$. Then we classify all polynomial maps of the form $H=(u(x,y,z),v(x,y,u), h(x,y))$ in the case that $JH$ is nilpotent and $(°v(x,y,0),°h)\leq 3$, $H(0)=0$. Finally, we classify polynomial maps of the form $H=(u(x,y,z),v(x,y,z), h(x,y))$ in certain conditions.
We study Poncelet's Theorem in finite projective coordinate planes over the field $GF(p)$ and concentrate on a particular pencil of conics. For pairs of such conics we investigate whether we can find polygons with $n$ sides, which are inscribed in one conic and circumscribed about the other, so-called Poncelet Polygons. By using suitable elements of the dihedral group for these pairs, we prove that the length $n$ of such Poncelet Polygons is independent of the starting point. In this sense Poncelet's Porism is valid. By using Euler's divisor sum formula for the totient function, we can make a …
Given an ideal $I$ in a Noetherian ring, one can ask the containment question: for which $m$ and $r$ is the symbolic power $I^{(m)}$ contained in the ordinary power $I^r$? C. Bocci and B. Harbourne study the containment question in a geometric setting, where the ideal $I$ is in a polynomial ring over a field. Like them, we will consider special geometric constructs. In particular, we obtain a complete solution in two extreme cases of ideals of points on a pair of lines in $\mathbb{P}^2$; in one case, the number of points on each line is the same, while in the other all the points but one are o…
We consider polynomial maps described by so-called "(multivariate) linearized polynomials". These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps without mixed terms over a characteristic zero field, we will only obtain (up to a linear transformation of the variables) triangular maps, which are the most basic examples of polynomial automorphisms. However, over the finite field F_q automorphisms defined by linearized polynomials have (in general) an entirely different structure. Namely, we will show tha…
We introduce a class of polynomial maps that we call polynomial roots of powerseries, and show that automorphisms with this property generate the automorphism group in any dimension. In particular we determine generically which polynomial maps that preserve the origin are roots of powerseries. We study the one-dimensional case in greater depth.
Given a rational monomial map, we consider the question of finding a toric variety on which it is algebraically stable. We give conditions for when such variety does or does not exist. We also obtain several precise estimates of the degree sequences of monomial maps on $¶^n$. Finally, we characterize polynomial maps which are algebraically stable on $(¶^1)^n$.
Let k be a field of characteristic zero. Let phi be a k-endomorphism of the polynomial algebra k[x_1,...,x_n]. It is known that phi is an automorphism if and only if it maps irreducible polynomials to irreducible polynomials. In this paper we show that phi satisfies the jacobian condition if and only if it maps irreducible polynomials to square-free polynomials. Therefore, the Jacobian Conjecture is equivalent to the following statement: every k-endomorphism of k[x_1,...,x_n], mapping irreducible polynomials to square-free polynomials, maps irreducible polynomials to irreducible polynomials.
Some properties of the relation between the singular point set and the non-proper value curve of polynomial maps of $\mathbb{C}^2$ are expressed in terms of Newton-Puiseux expansions.
In this paper, we give two algorithms to compute preimages of curves under polynomial endomorphisms. In particular, this gives an efficient way of computing preimages of points. Furthermore, we explain the abstract setting under which one can iteratively compute the inverse of a polynomial automorphism.
The aim of this note is to give an effective criterion to verify whether a cubic polynomial over a non-Archimedean field has a weak Néron model or not.
