## Abstract

Suppose that there exists a hypersurface with the Newton polytope Δ , which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ from below. As a particular application of our method we consider a planar algebraic curve C which passes through generic points p_{1}, … , p_{n} with prescribed multiplicities m_{1}, … , m_{n}. Suppose that the minimal lattice width ω(Δ) of the Newton polygon Δ of the curve C is at least max (m_{i}). Using tropical floor diagrams (a certain degeneration of p_{1}, … , p_{n} on a horizontal line) we prove that area(Δ)≥12∑i=1nmi2-S,whereS=12max(∑i=1nsi2|si≤mi,∑i=1nsi≤ω(Δ)).In the case m_{1}= m_{2}= ⋯ = m≤ ω(Δ) this estimate becomes area(Δ)≥12(n-ω(Δ)m)m2. That rewrites as d≥(n-12-12n)m for the curves of degree d. We consider an arbitrary toric surface (i.e. arbitrary Δ) and our ground field is an infinite field of any characteristic, or a finite field large enough. The latter constraint arises because it is not a priori clear what is a collection of generic points in the case of a small finite field. We construct such collections for fields big enough, and that may be also interesting for the coding theory.

Original language | English |
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Pages (from-to) | 158-179 |

Number of pages | 22 |

Journal | Discrete and Computational Geometry |

Volume | 58 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jul 2017 |

Externally published | Yes |

## Keywords

- Floor diagrams
- Nagata’s conjecture
- Tropical geometry
- m-Fold point