Descartes' Rule of Signs for Polynomial Systems Supported on Circuits
summary
Generalizes Descartes' rule to multivariate polynomial systems supported on circuits by giving an explicit procedure: form the Gale dual matrix from the support and coefficients, compute determinants of signed submatrices, and count sign variations in that determinant sequence to get an upper bound on positive real solutions. Proves the bound is tight by building constructive families attaining the bound; for a six-point circuit one example produces 10 positive solutions. Tools: elimination via Gale duality, determinant sign analysis and fewnomial combinatorics.
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