Abstract. In this paper, we establish some further results on tropical compactifications. We give an affirmative answer to a conjecture of Tevelev in characteristic 0: any variety contains a Schön very affine open subvariety. Also we show that any fan supported on the tropicalization of a Schön very affine variety produces a Schön compactification. As an application, we show that the moduli space of six points of P 2 in linear general position is Hübsch. Using toric schemes over a discrete valuation ring, we extend tropical compactifications to the nonconstant coefficient case.

Let $\mathbf{k}=(k_1,k_2,\cdots,k_t)$ be a $t$-tuple of integers, and $m$ be a positive integer. For a subset $A\subset\mathbf{Z}_m$ and any $n\in\mathbf{Z}_m$, let $r_A^{\mathbf{k}}(n)$ denote the number of solutions of the equation $k_1a_1+\cdots+k_ta_t=n$ with $a_1,\cdots,a_t\in A$. In this paper, we give a necessary and sufficient condition on $(\mathbf{k},m)$ such that there exists a subset $A\subset \mathbf{Z}_m$ satisifying $r_{A}^{\mathbf{k}}=r_{\mathbf{Z}_m\backslash A}^{\mathbf{k}}$. This settles a problem of Yang and Chen.

a b s t r a c t For a given positive integer N, and any coloring function c : N → {0, 1} satisfying c(2k) = 1 − c(k), c(2k + 1) = c(k) for all k ≥ N, we show that for all n ≥ 20N, n has both a monochromatic representation and a multicolored representation, in other words, there exist x, y, u, v ∈ N, such that n = x+y = u+v, c(x) = c(y) and c(u) ̸ = c(v). Similar results are obtained for another kind of coloring function c : N → {0, 1} satisfying c(2k) = c(k) and c(2k + 1) = 1 − c(k) for all k ≥ N. This answers a question of Y.-G. Chen on the values of representation functions.

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