Jens Forsgård

Could René Descartes Have Known This

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

Lopsided approximation of amoebas

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

Coamoebas of Polynomials Supported on Circuits

We study coamoebas of polynomials supported on circuits. Our results include an explicit description of the space of coamoebas, a relation between connected components of the coamoeba complement and critical points of the polynomial, an upper bound on the area of a planar coamoeba, and a recovered bound on the number of positive solutions of a fewnomial system.

Defective dual varieties for real spectra

We introduce an invariant of a finite point configuration

On the order map for hypersurface coamoebas

A \subset \mathbb {R}^{1+n}

Euler-Mellin integrals and A-hypergeometric functions

A⊂R1+n which we denote the cuspidal form of A. We use this invariant to extend Esterov’s characterization of dual-defective point configurations to exponential sums; the dual variety associated with A has codimension at least 2 if and only if A does not contain any iterated circuit.

Coamoebas and line arrangements in dimension two

ABSTRACT Here we provide a correct version of Proposition 6 of [FKS]. No other results of the latter paper are affected.