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A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $$\mathbb{Z}_2$$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that...
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