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- W1541058864 abstract "Suppose that there exists a hypersurface with the Newton polytope $$Delta $$ , which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of $$Delta $$ to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of $$Delta $$ from below. As a particular application of our method we consider a planar algebraic curve C which passes through generic points $$p_1,ldots ,p_n$$ with prescribed multiplicities $$m_1,ldots ,m_n$$ . Suppose that the minimal lattice width $$omega (Delta )$$ of the Newton polygon $$Delta $$ of the curve C is at least $$max (m_i)$$ . Using tropical floor diagrams (a certain degeneration of $$p_1,ldots , p_n$$ on a horizontal line) we prove that $$begin{aligned} {{mathrm {area}}}(Delta )ge & {} frac{1}{2}sum _{i=1}^n m_i^2-S, text {where } S= & {} frac{1}{2}max left( sum _{i=1}^n s_i^2, Big |, s_ile m_i, sum _{i=1}^n s_ile omega (Delta )right) . end{aligned}$$ In the case $$m_1=m_2=cdots =mle omega (Delta )$$ this estimate becomes $${mathrm {area}}(Delta )ge frac{1}{2}bigl (n-frac{omega (Delta )}{m}bigr )m^2$$ . That rewrites as $$dge bigl (sqrt{n}-frac{1}{2}-frac{1}{2sqrt{n}}bigr )m$$ for the curves of degree d. We consider an arbitrary toric surface (i.e. arbitrary $$Delta $$ ) 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." @default.
- W1541058864 created "2016-06-24" @default.
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- W1541058864 date "2017-05-11" @default.
- W1541058864 modified "2023-10-03" @default.
- W1541058864 title "Tropical Approach to Nagata’s Conjecture in Positive Characteristic" @default.
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- W1541058864 doi "https://doi.org/10.1007/s00454-017-9894-7" @default.
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