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Charles Arnal

Publications and source records attributed to Charles Arnal.

23 records · Page 2Linked to original sources
arXiv2023

Coefficients of almost-degenerate density matrix perturbation theory for eigenvalue problems

We investigate almost-degenerate perturbation theory of eigenvalue problems, using spectral projectors, also named density matrices. When several eigenvalues are close to each other, the coefficients of the perturbative series become singular because inverses of differences between eigenvalues arise as some factors. We remove those artificial singularities in the expressions of the coefficients of the series, allowing eigenvalue gaps to be arbitrarily small and even vanishing in the resulting formulas.

math-ph↗
arXiv2022

Patchworking real algebraic hypersurfaces with asymptotically large Betti numbers

In this article, we describe a recursive method for constructing a family of real projective algebraic hypersurfaces in ambient dimension $n$ from families of such hypersurfaces in ambient dimensions $k=1,\ldots,n-1$. The asymptotic Betti numbers of real parts of the resulting family can then be described in terms of the asymptotic Betti numbers of the real parts of the families used as ingredients. The algorithm is based on Viro's Patchwork and inspired by I. Itenberg's and O. Viro's construction of asymptotically maximal families in arbitrary dimension. Using it, we prove that for any $n$ and $i=0,\ldots,n-1$, there is a family of asymptotically maximal real projective algebraic hypersurfaces $\{Y^n_d\}_d$ in $\mathbb{R} \mathbb{P} ^n$ (where $d$ denotes the degree of $Y^n_d$) such that the $i$-th Betti numbers $b_i(\mathbb{R} Y^n_d)$ are asymptotically strictly greater than the $(i,n-1-i)$-th Hodge numbers $h^{i,n-1-i}(\mathbb{C} Y^n_d)$. We also build families of real projective algebraic hypersurfaces whose real parts have asymptotic (in the degree $d$) Betti numbers that are asymptotically (in the ambient dimension $n$) very large.

math.AG↗
arXiv2021

Complex conjugation and simplicial algebraic hypersurfaces

We call a real algebraic hypersurface in $(\mathbb{C}^*)^n$ simplicial if it is given by a real Laurent polynomial in $n$-variables that has exactly $n+1$ monomials with non-zero coefficients and such that the convex hull in $\mathbb{R}^n$ of the $n+1$ points of $\mathbb{Z} ^n$ corresponding to the exponents is a non-degenerate $n$-dimensional simplex. Such hypersurfaces are natural building blocks from which more complicated objects can be constructed, for example using O. Viro's Patchworking method. Drawing inspiration from related work by G. Kerr and I. Zharkov, we describe the action of the complex conjugation on the homology of the coamoebas of simplicial real algebraic hypersurfaces, hoping it might prove useful in a variety of problems related to topology of real algebraic varieties. In particular, assuming a reasonable conjecture, we identify the conditions under which such a hypersurface is Galois maximal.

math.AG↗
arXiv2019

Lefschetz section theorems for tropical hypersurfaces

We establish variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of tropical hypersurfaces of toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which are compact or contained in $\mathbb{R}^n$ are torsion free. We prove a relationship between the coefficients of the $χ_y$ genera of complex hypersurfaces in toric varieties and Euler characteristics of the integral tropical cellular chain complexes of their tropical counterparts. It follows that the integral tropical homology groups give the Hodge numbers of compact non-singular hypersurfaces of complex toric varieties. Finally for tropical hypersurfaces in certain affine toric varieties, we relate the ranks of their tropical homology groups to the Hodge-Deligne numbers of their complex counterparts.

math.AG↗