# Tropical secant graphs of monomial curves

Abstract : We construct and study an embedded weighted balanced graph in $\mathbb{R}^{n+1}$ parametrized by a strictly increasing sequence of $n$ coprime numbers $\{ i_1, \ldots, i_n\}$, called the $\textit{tropical secant surface graph}$. We identify it with the tropicalization of a surface in $\mathbb{C}^{n+1}$ parametrized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector $(0, i_1, \ldots, i_n)$, which can be described by a balanced graph called the $\textit{tropical secant graph}$. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in $\mathbb{P}^4$.
Keywords :
Document type :
Conference papers
Domain :

Cited literature [10 references]

https://hal.inria.fr/hal-01186248
Contributor : Coordination Episciences Iam Connect in order to contact the contributor
Submitted on : Monday, August 24, 2015 - 3:44:51 PM
Last modification on : Wednesday, August 7, 2019 - 12:19:22 PM
Long-term archiving on: : Wednesday, November 25, 2015 - 4:46:32 PM

### File

dmAN0147.pdf
Publisher files allowed on an open archive

### Citation

María Angélica Cueto, Shaowei Lin. Tropical secant graphs of monomial curves. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.669-680, ⟨10.46298/dmtcs.2820⟩. ⟨hal-01186248⟩

Record views