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A perfect power is a positive integer of the form a x where a ≥ 1 and x ≥ 2 are rational integers. Subbayya Sivasankaranarayana Pillai wrote several papers on these numbers. In 1936 and again in 1945 he suggested that for any given k ≥ 1, the number of positive integer solutions (a, b, x, y), with x ≥ 2 and y ≥ 2, to the Diophantine equation a x − b y = k… (More)

A theorem of Kušnirenko and Bernštein shows that the number of isolated roots of a system of polynomials in a torus is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of… (More)

- Patrice Philippon
- J. London Math. Society
- 2015

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