13 citations · 32 across the 22 of their papers we have counts for
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Beyond Harnack Amoeba Boundaries
Mounir Nisse
We construct smooth irreducible plane curves whose complete amoeba contour is a smooth embedded curve equal to the amoeba boundary, although the curves are not torically equivalent…
Beyond Harnack Rigidity
Mounir Nisse
The boundary of a plane amoeba is always contained in its contour, and equality is a characteristic feature of simple Harnack curves. We show that the converse fails, even under st…
On the Analyticity of Amoeba Contours
Mounir Nisse
The contour of an amoeba records the critical values of the logarithmic map and forms a fundamental interface between complex algebraic geometry and real-analytic geometry. Buildin…
Cusps and nodes of Amoeba Contours
Mounir Nisse
We establish new Newton-polygon bounds for the singularities of amoeba contours of smooth plane curves. Our cusp estimate refines the degree-four bound of Lang--Shapiro--Shustin, r…
Singularities of Amoeba Contours
Mounir Nisse
We give explicit real-algebraic equations for computing the singularities of amoeba contours, separating degeneracies of the logarithmic critical locus from coincidences of distinc…
Sparse Bounds for Amoeba Contours
Mounir Nisse
We derive new upper bounds for the real degree of the contour of the amoeba of algebraic curves and hypersurfaces. Our approach refines the Pfaffian method of Lang--Shapiro--Shusti…
Contour Degree of Amoebas of Complete Intersections
Mounir Nisse
We establish the first universal upper bounds for the real degree of the contour of the amoeba of a smooth complete intersection in the algebraic torus. Our approach extends the Pf…
Phase limit sets of linear spaces and discriminants
Mounir Nisse, Frank Sottile
We show that the closure of the coamoeba of a linear space/hyperplane complement is the union of products of coamoebas of hyperplane complements coming from flags of flats, and rel…
Links represented by phases of algebraic curves
Yen-Kheng Lim, Mounir Nisse
The prime motivation behind this paper is to prove that any torus link can be realized as the union of the one-dimensional connected components of the set of critical values of the…
New Subexponential Fewnomial Hypersurface Bounds
Jens Forsgård, Mounir Nisse, J. Maurice Rojas
Suppose are real numbers, is a set of points not all lying in the same affine hyperplane, $y\!\in\!\mathbb{R}^n…
Amoeba-shaped polyhedral complex of an algebraic hypersurface
Mounir Nisse, Timur Sadykov
Given a complex algebraic hypersurface~, we introduce a polyhedral complex which is a subset of the Newton polytope of the defining polynomial for~ and enjoys the key topolog…
Geometry and a natural symplectic structure of phase tropical hypersurfaces
Young Rock Kim, Mounir Nisse
First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in . Next, we prove that complex hyperplan…
Higher convexity for complements of tropical varieties
Mounir Nisse, Frank Sottile
We consider Gromov's homological higher convexity for complements of tropical varieties, establishing it for complements of tropical hypersurfaces and curves, and for nonarchimedea…
A tropical characterization of complex analytic varieties to be algebraic
Farid Madani, Lamine Nisse, Mounir Nisse
In this paper we study a -dimensional analytic subvariety of the complex algebraic torus. We show that if its logarithmic limit set is a finite rational -dimensional sphe…
Higher convexity of coamoeba complements
Mounir Nisse, Frank Sottile
We show that the complement of the coamoeba of a variety of codimension k+1 is k-convex, in the sense of Gromov and Henriques. This generalizes a result of Nisse for hypersurface c…
Amoeba finite basis of solutions to a system of polynomials in variables
Mounir Nisse
We show that the amoeba of a complex algebraic variety defined as the solutions to a generic system of polynomials in variables has a finite basis. In other words, it is th…
Amoeba finite basis does not exist in general
Mounir Nisse
We show that the amoeba of a generic complex algebraic variety of codimension do not have a finite basis. In other words, it is not the intersection of finitely many hypers…
Metric Estimates and Membership Complexity for Archimedean Amoebae and Tropical Hypersurfaces
Martin Avendano, Roman Kogan, Mounir Nisse +1
Given any complex Laurent polynomial , is the image of its complex zero set under the coordinate-wise log absolute value map. We give an efficiently constru…
Generalized logarithmic Gauss map and its relation to (co)amoebas
Farid Madani, Mounir Nisse
We define the generalized logarithmic Gauss map for algebraic varieties of the complex algebraic torus of any codimension. Moreover, we describe the set of critical points of the l…
Amoebas and coamoebas of linear spaces
Mounir Nisse, Mikael Passare
We give a complete description of amoebas and coamoebas of -dimensional very affine linear spaces in . This include an upper bound of their dimension, and we…
Analytic varieties with finite volume amoebas are algebraic
Farid Madani, Mounir Nisse
In this paper, we study the amoeba volume of a given dimensional generic analytic variety of the complex algebraic torus $(\C^*)^n$. When , we show that is al…
The Phase Limit Set of a Variety
Mounir Nisse, Frank Sottile
A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. We describe the structure of the boundary of the coamoeba of a variety, which w…
On the Volume of Complex Amoebas
Farid Madani, Mounir Nisse
The paper deals with amoebas of -dimensional algebraic varieties in the algebraic complex torus of dimension . First, we show that the area of complex algebraic curve…
Geometric and Combinatorial Structure of Hypersurface Coamoebas
Mounir Nisse
Let be a complex algebraic hypersurface defined by a polynomial with Newton polytope . It is well known that the spine of its amoeba has a structure of a tropical hypers…