Journal•ISSN: 1615-715X
Advances in Geometry
De Gruyter
About: Advances in Geometry is an academic journal published by De Gruyter. The journal publishes majorly in the area(s): Curvature & Polytope. It has an ISSN identifier of 1615-715X. Over the lifetime, 753 publications have been published receiving 6925 citations.
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TL;DR: The tropical Grassmannian G2; n as mentioned in this paper is a simplicial complex glued from 1035 tetrahedra, and it is a polyhedral subcomplex of the Grobner fan.
Abstract: In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Grobner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plucker relations. It parametrizes all tropical linear spaces. Lines in tropical projective space are trees, and their tropical Grassmannian G2; n equals the space of phylogenetic trees studied by Billera, Holmes and Vogtmann. Higher Grassmannians oer a natural generalization of the space of trees. Their faces correspond to monomial-free initial ideals of the Plucker ideal. The tropical Grassmannian G3; 6 is a simplicial complex glued from 1035 tetrahedra.
562 citations
TL;DR: For odd n ≥ 3, the moment problem for K is finitely solvable if K is bounded on two half-branches of C(R) at infinity which represent different points at infinity as mentioned in this paper.
Abstract: Example 4: The assertion for n ≤ 2 is true. Also, for odd n ≥ 3, the assertion is true, since in this case the curve C has exactly one point at infinity, which is real. If n ≥ 4 is even (and C is real), there are two points at infinity which are both real. Therefore, the moment problem for K is not finitely solvable if K is unbounded on two half-branches of C(R) at infinity which represent different points at infinity. Otherwise, the moment problem for K is finitely solvable. More concretely, this means the following for n ≥ 4. Let Q1, Q2, Q3, Q4 be the four quadrants of the real plane (numbered counter-clockwise in the usual way). If n ≡ 0 (mod 4), the moment problem for K is finitely solvable iff at least one of K ∩ (Q1 ∪Q2), K ∩ (Q3 ∪Q4) is bounded. If n ≡ 2 (mod 4), the moment problem for K is finitely solvable iff at least one of K ∩ (Q1 ∪Q3), K ∩ (Q2 ∪Q4) is bounded.
112 citations
TL;DR: In this paper, it was shown that every continuous rational map between unit spheres is homotopic to a rational map of such a type, and connections with algebraic cycles and vector bundles were established.
Abstract: The paper deals with rational maps between real algebraic sets. We are interested in the rational maps which extend to continuous maps defined on the entire source space. In particular, we prove that every continuous map between unit spheres is homotopic to a rational map of such a type. We also establish connections with algebraic cycles and vector bundles.
77 citations
TL;DR: In this article, a family of algebraic curves over the odd number of n is introduced, and it is shown that these curves are maximal when n = 3, which is the first maximal curve proven not to be covered by the Hermitians.
Abstract: We introduce a family of algebraic curves over $\F_{q^{2n}}$ (for an odd $n$) and show that they are maximal. When $n=3$, our curve coincides with the $\F_{q^6}$-maximal curve that has been found by Giulietti and Korchm\'{a}ros recently. Their curve (i.e., the case $n=3$) is the first example of a maximal curve proven not to be covered by the Hermitian curve.
75 citations
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TL;DR: In this article, a convex polygon P in the projective plane can be formed by taking the pairwise intersections of the lines extending the edges of P. When P is a Poncelet polygon, it is shown that this grid is contained in a finite union of ellipses and hyperbolas.
Abstract: Given a convex polygon P in the projective plane we can form a finite “grid” of points by taking the pairwise intersections of the lines extending the edges of P . When P is a Poncelet polygon we show that this grid is contained in a finite union of ellipses and hyperbolas and derive other related geometric information about the grid.
63 citations