Frédéric Bihan

Faster real feasibility via circuit discriminants

We show that detecting real roots for honestly <i>n</i>-variate (<i>n</i>+2)-nomials with integer exponents and coefficients) can be done in time polynomial in the sparse encoding for any fixed n<i>n</i>. The best previous complexity bounds were exponential in the sparse encoding, even for <i>n</i> fixed. We then give a characterization of those functions <i>k</i>(<i>n</i>) such that the complexity of detecting real roots for <i>n</i>-variate(<i>n</i>+<i>k</i>(<i>n</i>))-nomials transitions from <b>P</b> to <b>NP</b>-hardness as <i>n</i> → ∞. Our proofs follow in large part from a new complexity threshold for deciding the vanishing of <i>A</i>-discriminants of <i>n</i>-variate (<i>n</i>+<i>k</i>(<i>n</i>))-nomials. Diophantine approximation, through linear forms in logarithms, also arises as a key tool.

Irrational Mixed Decomposition and Sharp Fewnomial Bounds for Tropical Polynomial Systems

Given convex polytopes

On the sharpness of fewnomial bounds and the number of components of fewnomial hypersurfaces

P_1,\ldots ,P_r \subset \mathbb {R}^n

Polynomial systems with few real zeroes

P1,…,Pr⊂Rn and finite subsets

GALE DUALITY FOR COMPLETE INTERSECTIONS

\mathcal {W}_I

Is Every Toric Variety an M-Variety?

WI of the Minkowski sums