Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains

Let $R$ be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let $L$ and $k$ be respectively its fraction field and residue field. Let $\Omega_R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of \Spec R. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to $\Omega_R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A[y]$, where $A$ is a complete discrete valuation ring with a not too restrictive condition on the residue field $k$, which is satisfied when $k$ is $C_1$.Comment: 11 pages, an argument in the proof of Lemma 5.1 is simplified; to appear in Annales de l'Institut Fourie

The Pythagoras number and the uu-invariant of Laurent series fields in several variables

We show that every sum of squares in the three-variable Laurent series field $\mathbb{R}((x,y,z))$ is a sum of 4 squares, as was conjectured in a paper of Choi, Dai, Lam and Reznick in the 1980's. We obtain this result by proving that every sum of squares in a finite extension of $\mathbb{R}((x,y))$ is a sum of $3$ squares. It was already shown in Choi, Dai, Lam and Reznick's paper that every sum of squares in $\mathbb{R}((x,y))$ itself is a sum of two squares. We give a generalization of this result where $\mathbb{R}$ is replaced by an arbitrary real field. Our methods yield similar results about the $u$-invariant of fields of the same type.Comment: final version, major revisions in the style of writing (abstract and introduction rewritten) compared to v.

Divisors of the Euler and Carmichael functions

We study the distribution of divisors of Euler's totient function and Carmichael's function. In particular, we estimate how often the values of these functions have "dense" divisors.Comment: v.3, 11 pages. To appear in Acta Arithmetica. Very small corrections and changes suggested by the referee. Added abstract, keywords, MS

Relative Severi inequality for fibrations of maximal Albanese dimension over curves

Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety $X$ of dimension $n \ge 2$ to a curve $B$ defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! \chi_f$, which was conjectured by Barja in [2]. Via the strategy outlined in [5], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $\chi_f > 0$, we prove that the general fiber $F$ of $f$ has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! \chi(F, \omega_F)$. We also prove some sharper results of the same type under extra assumptions.Comment: Comments are welcom