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 p₁, … , p_n with prescribed multiplicities m₁, … , 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 p₁, … , 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 m₁= m₂= ⋯ = 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.