arXiv subjects
Publications and source records attributed to Benoit Bertrand.
A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision. It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a coherent way we prove that no convexity hypothesis is required to patchwork such curves.
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 for any polytope $Δ$ there are families of hypersurfaces with the Newton polytopes $(λΔ)_{λ\in \mathbb{N}}$ that are asymptotically maximal when $λ$ tends to infinity. We also show that these results generalize to complete intersections.
We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii bounds for these systems. When the support is exactly a circuit whose affine span is ${\Z}^n$, this bound is $2n+1$, while the Khovanskii bound is exponential in $n^2$. The bound $2n+1$ can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic.