arXiv subjects

David A. Madore

Publications and source records attributed to David A. Madore.

4 recordsLinked to original sources

arXiv2015

The Hadwiger-Nelson problem over certain fields

We compute the Hadwiger-Nelson numbers $χ(E^2)$ for certain number fields $E$, that is, the smallest number of colors required to color the points in the plane with coordinates in~$E$ so that no two points at distance $1$ from one another have the same color. Specifically, we show that $χ(\mathbb{Q}(\sqrt{2})^2) = 2$, that $χ(\mathbb{Q}(\sqrt{3})^2) = 3$, that $χ(\mathbb{Q}(\sqrt{7})^2) = 3$ despite the fact that the graph $Γ(\mathbb{Q}(\sqrt{7})^2)$ is triangle-free, and that $4 \leq χ(\mathbb{Q}(\sqrt{3}, \sqrt{11})^2) \leq 5$. We also discuss some results over other fields, for other quadratic fields. We conclude with some comments on the use of the axiom of choice.

math.CO↗

arXiv2014

Calculabilité de la cohomologie étale modulo l

Let $X$ be an algebraic scheme over an algebraically closed field and $\ell$ a prime number invertible on $X$. According to classical results (due essentially to A. Grothendieck, M. Artin and P. Deligne), the étale cohomology groups $\mathrm{H}^i(X,\mathbb{Z}/\ell\mathbb{Z})$ are finite-dimensional. Using an $\ell$-adic variant of M. Artin's good neighborhoods and elementary results on the cohomology of pro-$\ell$ groups, we express the cohomology of $X$ as a well controlled colimit of that of toposes constructed on $BG$ where the $G$ are computable finite $\ell$-groups. From this, we deduce that the Betti numbers modulo $\ell$ of $X$ are algorithmically computable (in the sense of Church-Turing). The proof of this fact, along with certain related results, occupies the first part of this paper. This relies on the tools collected in the second part, which deals with computational algebraic geometry. Finally, in the third part, we present a "universal" formalism for computation on the elements of a field.

math.AG↗

arXiv2006

Les hypersurfaces cubiques sont séparablement rationnellement connexes

This note (which makes no claim to novelty) presents a proof of the separable rational connectedness of smooth cubic hypersurfaces, in any characteristic, by showing how to explicitly construct very free curves (of degree 3) on them. ----- Cette note (qui ne prétend pas à l'originalité) démontre la séparable rationnelle connexité des hypersurfaces cubiques lisses, en toute caractéristique, en construisant explicitement des courbes très libres (de degré 3) tracées dessus.

math.AG↗

arXiv2003

Surfaces de del Pezzo sans point rationnel sur un corps de dimension cohomologique un

For each integer d=2,3,4, there exists a field F with cohomological dimension 1 and a del Pezzo surface of degree d over F having no rational point. Proofs use the theorem of Merkur'ev and Suslin, the Riemann-Roch theorem on a surface and Rost's degree formula. ----- Pour chaque entier d=2,3,4, il existe un corps F de dimension cohomologique 1 et une surface de del Pezzo de degre d sur F sans point rationnel. Les demonstrations utilisent le theoreme de Merkur'ev et Suslin, le theoreme de Riemann-Roch sur une surface et la formule du degre de Rost.

math.NT↗