Rational points of fixed denominator in real toric arrangements

We give a sufficient condition on a positive integer $m$ for every stratum of a given real toric hyperplane arrangement to contain a rational point of denominator $m$. As a consequence, we give a sufficient condition on $m$ for the degree $m$ Frobenius pushforward of the structure sheaf on a smooth toric variety to contain all possible summands in the Picard group.

💡 Research Summary

The paper studies rational points of a fixed denominator in the strata of a real toric hyperplane arrangement and derives a sufficient arithmetic condition guaranteeing that every stratum contains such a point.
Let (A={v_1,\dots ,v_k}\subset\mathbb Z^n) be a finite set of integer vectors. For each (v_i) define a circle‑valued function on the real torus (\mathbb T^n=\mathbb R^n/\mathbb Z^n) by (f_i(x)=v_i\cdot x). The zero‑level sets (h_i=f_i^{-1}(0)) are subtori, and the collection ({h_i}) determines a stratification (S_A) of (\mathbb T^n) (the locally closed pieces obtained by intersecting some of the (h_i) and removing the others).

For a positive integer (m) let
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