Maximally Sparse Polynomials have Solid Amoebas

2007/04/17 by Mounir Nisse, Nisse, Mounir · 2 citations
Computer Science · Mathematics · #Polynomial and algebraic computation #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.0704.2216

Abstract

Let f be an ordinary polynomial in ℂ[z₁,..., z_n] with no negative exponents and with no factor of the form z₁^α₁... z_n^α_n where α_i are non zero natural integer. If we assume in addicting that f is maximally sparse polynomial (that its support is equal to the set of vertices of its Newton polytope), then a complement component of the amoeba \mathscrA_f in ℝⁿ of the algebraic hypersurface V_f⊂ (ℂ^*)ⁿ defined by f, has order lying in the support of f, which means that \mathscrA_f is solid. This gives an affirmative answer to Passare and Rullgård question in [PR2-01].

Citations

• Laurent Determinants and Arrangements of Hyperplane Amoebas

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• Coamoebas of complex algebraic plane curves and the logarithmic Gauss map
• Geometric and Combinatorial Structure of Hypersurface Coamoebas

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