In this article, we explore the connections between nonnegativity, theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, cover boundary sonc-cone by semialgebraic sets that are parametrized families hypersurfaces. As application, give sufficient conditions for equality sparse nonnegativity cone generic sets, describe a stratification in univariate case.

Abstract Consider a sparse system of Laurent polynomials in variables with complex coefficients and support finite lattice set . The maximal number isolated roots the torus is known to be normalized volume convex hull (the BKK bound). We explore following question: if cardinality equals , what maximum local intersection multiplicity at one point terms ? This study was initiated by Gabrielov [13] multivariate case. give an upper bound that always sharp when and, under technical hypothesis, it...

We consider integrals that generalize both Mellin transforms of rational functions the form 1/f and classical Euler integrals. The domains integration our so-called are naturally related to coamoeba f, components complement closure this give rise a family these After performing an explicit meromorphic continuation integrals, we interpret them as A-hypergeometric discuss their linear independence relation Barnes

Below we discuss the partition of space real univariate polynomials according to number positive and negative roots signs coefficients. We present several series non-realizable combinations together with numbers roots. provide a detailed information about possible up degree 8 as well general conjecture such combinations.

Given a hypersurface coamoeba of Laurent polynomial f, it is an open problem to describe the structure set connected components its complement. In this paper we approach by introducing lopsided coamoeba. We show that closed comes naturally equipped with order map, i.e. map from complement translated lattice inside zonotope Gale dual point configuration $\operatorname{supp}(f)$. Under natural assumption, bijection. Finally use obtain new results concerning coamoebas polynomials small codimension.

We prove that for any degree |$d$|⁠, there exist (families of) finite sequences |$\{\lambda_{k,d}\}_{0\le k\le d}$| of positive numbers such that, real polynomial |$P$| the number its roots is less than or equal to so-called essential tropical obtained from by multiplication coefficients |$\lambda_{0,d},\lambda_{1,d},\dots , \lambda_{d,d}$|⁠, respectively. In particular, univariate |$P(x)$| |$d$| with a non-vanishing constant term, we conjecture one can take |$\lambda_{k,d}={\rm...

The amoeba of a Laurent polynomial is the image corresponding hypersurface under coordinatewise log absolute value map. In this article, we demonstrate that theoretical approximation method due to Purbhoo can be used efficiently in practice. To do this, resolve main bottleneck Purbhoo's by exploiting relations between cyclic resultants. We use same approach give an Log preimage using semi-algebraic sets. also provide SINGULAR/Sage implementation these algorithms, which shows significant...

We describe the relationship between dimer models on real two-torus and coamoebas of curves in (\mathbb C^\times)^2 . show, inter alia, that model obtained from shell coamoeba is a deformation retract closed if only number connected components complement maximal. Furthermore, we show general characteristic polynomial does not have maximal its

We introduce an invariant of a finite point configuration $$A \subset \mathbb {R}^{1+n}$$ which we denote the cuspidal form A. use this to extend Esterov’s characterization dual-defective configurations exponential sums; dual variety associated with A has codimension at least 2 if and only does not contain any iterated circuit.

We describe the parametric behavior of series solutions an $A$-hypergeometric system. More precisely, we construct explicit stratifications parameter space such that, on each stratum, system are holomorphic.

We prove that for any degree d, there exist (families of) finite sequences a_0, a_1,..., a_d of positive numbers such that, real polynomial P the number its roots is less than or equal to so-called essential tropical obtained from by multiplication coefficients a_1,... respectively. In particular, univariate d with non-vanishing constant term, we conjecture one can take a_k = e^{-k^2}, k 0, ... , d. The latter claim be thought as a generalization Descartes's rule signs. settle this up 4 well...

Abstract In this paper we explore special values of Gaussian hypergeometric functions in terms products Euler $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> -functions and exponential linear the parameters. They include some classical evaluations, but main inspiration is from contiguity method recently applied by Akihito Ebisu.

Consider a sparse system of n Laurent polynomials in variables with complex coefficients and support finite lattice set A. The maximal number isolated roots the n-torus is known to be normalized volume convex hull A (the BKK bound). We explore following question: if cardinality equals n+m+1, what maximum local intersection multiplicity at one point torus terms m? This study was initiated by Gabrielov multivariate case. give an upper bound that always sharp when m=1 and, under generic...

In this article, we explore the connections between nonnegativity, theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, cover boundary sonc-cone by semi-algebraic sets that are parametrized families hypersurfaces. As application, characterization generic for which is equal to sparse nonnegativity cone, describe a stratification in univariate case.

We consider integrals that generalize both the Mellin transforms of rational functions form 1/f and classical Euler integrals. The domains integration our so-called Euler--Mellin are naturally related to coamoeba f, components complement closure give rise a family these After performing an explicit meromorphic continuation integrals, we interpret them as A-hypergeometric discuss their linear independence relation Mellin--Barnes