Describing amoebas

Computing Amoebas

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 Purbhoo’s method by exploiting relations between cyclic resultants. We use the same appro...

Amoebas and Sdp

Abstract. The hypersurface amoeba membership problem is the problem to decide whether for an input point λ ∈ R, the point λ is an element of the amoeba of the hypersurface generated by a multivariate complex Laurent polynomial c. We present an algorithm to solve this problem based on semidefinite programming and on Lasserre’s moment method for approximating the minimum of a real polynomial. The...

A Nullstellensatz for amoebas

The amoeba of an affine algebraic variety V ⊂ (C∗)r is the image of V under the map (z1, . . . , zr) 7→ (log |z1|, . . . , log |zr|). We give a characterisation of the amoeba based on the triangle inequality, which we call ‘testing for lopsidedness’. We show that if a point is outside the amoeba of V , there is an element of the defining ideal which witnesses this fact by being lopsided. This c...

Differential Equations for Morphological Amoebas

This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We investigate ...