R+ˆ is surprisingly an integral domain

It is shown that if $(R,\mathfrak{m},k)$ is a complete local domain with $\mathrm{char}k=p>0$ and $R^{+}$ is its integral closure in an algebraic closure of the quotient field, then both the $\mathfrak{m}$-adic and p-adic completions of $R^{+}$ are integral domains. More generally, this theorem remains true if the completeness assumption is relaxed to allow R to be an analytically irreducible Henselian local ring. It is also shown that these rings, which are Cohen-Macaulay R-modules (even balanced in the $\mathfrak{m}$-adic case), will have dimension larger than the dimension of R unless $\dim ?R\le 1$.     

Projective well orders and coanalytic witnesses

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which $\mathfrak{a}=\mathfrak{u}=\mathfrak{i}={\mathrm{?}}_1<2^{{\mathrm{?}}_0}={\mathrm{?}}_2$, each of $\mathfrak{a}$, $\mathfrak{u}$, $\mathfrak{i}$ has a ${\mathrm{\Pi }}_1^1$ witness and there is a ${\mathrm{\Delta }}_3^1$ well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a ${\mathrm{\Delta }}_3^1$ well-order of the reals is consistent with $\mathfrak{c}={\mathrm{?}}_2$ and each of the following: $\mathfrak{a}=\mathfrak{u}<\mathfrak{i}$, $\mathfrak{a}=\mathfrak{i}<\mathfrak{u}$, $\mathfrak{a}<\mathfrak{u}=\mathfrak{i}$, where the smaller cardinal characteristics have co-analytic witnesses.Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type.     

On the Newton polygons of twisted L-functions of binomials

Let χ be an order c multiplicative character of a finite field and $f(x)=x^d+\lambda x^e$ a binomial with $(d,e)=1$. We study the twisted classical and T-adic Newton polygons of f. When $p>(d-e)(2d-1)$, we give a lower bound of Newton polygons and show that they coincide if p does not divide a certain integral constant depending on $p\mathrm{mod}cd$.We conjecture that this condition holds if p is large enough with respect to $c,d$ by combining all known results and the conjecture given by Zhang-Niu. As an example, we show that it holds for $e=d-1$.     

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.