Suppose that there exists a hypersurface with the Newton polytope Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document}, which passes through a given set of subvarieties. Using tropical geometry, we associate a subset of Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} to each of these subvarieties. We prove that a weighted sum of the volumes of these subsets estimates the volume of Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} from below. As a particular application of our method we consider a planar algebraic curve C which passes through generic points p1,…,pn\documentclass[12pt]{minimal}
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\begin{document}$$p_1,\ldots ,p_n$$\end{document} with prescribed multiplicities m1,…,mn\documentclass[12pt]{minimal}
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\begin{document}$$m_1,\ldots ,m_n$$\end{document}. Suppose that the minimal lattice width ω(Δ)\documentclass[12pt]{minimal}
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\begin{document}$$\omega (\Delta )$$\end{document} of the Newton polygon Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} of the curve C is at least max(mi)\documentclass[12pt]{minimal}
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\begin{document}$$\max (m_i)$$\end{document}. Using tropical floor diagrams (a certain degeneration of p1,…,pn\documentclass[12pt]{minimal}
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\begin{document}$$p_1,\ldots , p_n$$\end{document} on a horizontal line) we prove that area(Δ)≥12∑i=1nmi2-S,whereS=12max∑i=1nsi2|si≤mi,∑i=1nsi≤ω(Δ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {{\mathrm {area}}}(\Delta )\ge & {} \frac{1}{2}\sum _{i=1}^n m_i^2-S,\ \ \text {where } \\ S= & {} \frac{1}{2}\max \left( \sum _{i=1}^n s_i^2\, \Big |\, s_i\le m_i, \sum _{i=1}^n s_i\le \omega (\Delta )\right) . \end{aligned}$$\end{document}In the case m1=m2=⋯=m≤ω(Δ)\documentclass[12pt]{minimal}
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\begin{document}$$m_1=m_2=\cdots =m\le \omega (\Delta )$$\end{document} this estimate becomes area(Δ)≥12(n-ω(Δ)m)m2\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {area}}(\Delta )\ge \frac{1}{2}\bigl (n-\frac{\omega (\Delta )}{m}\bigr )m^2$$\end{document}. That rewrites as d≥(n-12-12n)m\documentclass[12pt]{minimal}
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\begin{document}$$d\ge \bigl (\sqrt{n}-\frac{1}{2}-\frac{1}{2\sqrt{n}}\bigr )m$$\end{document} for the curves of degree d. We consider an arbitrary toric surface (i.e. arbitrary Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document}) 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.
