Families of artinian and low dimensional determinantal rings

Let $\mathrm{GradAlg}(H)$ be the scheme parameterizing graded quotients of $R=k[x_0,\dots ,x_n]$ with Hilbert function H (it is a subscheme of the Hilbert scheme of ${\mathbb{P}}^n$ if we restrict to quotients of positive dimension, see definition below). A graded quotient $A=R/I$ of codimension c is called standard determinantal if the ideal I can be generated by the $t\times t$ minors of a homogeneous $t\times (t+c?1)$ matrix $(f_{ij})$. Given integers $a_0\le a_1\le ...\le a_{t+c?2}$ and $b_1\le ...\le b_t$, we denote by $W_s(\underset{\_}{b};\underset{\_}{a})?\mathrm{GradAlg}(H)$ the stratum of determinantal rings where $f_{ij}\in R$ are homogeneous of degrees $a_j?b_i$.In this paper we extend previous results on the dimension and codimension of $W_s(\underset{\_}{b};\underset{\_}{a})$ in $\mathrm{GradAlg}(H)$ to artinian determinantal rings, and we show that $\mathrm{GradAlg}(H)$ is generically smooth along $W_s(\underset{\_}{b};\underset{\_}{a})$ under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of ${\mathbb{P}}^n$ is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of $\mathrm{GradAlg}(H)$.     

Sampling in a weighted Sobolev space

We show that functions f in some weighted Sobolev space are completely determined by time-frequency samples ${\{f(t_n)\}}_{n\in \mathbb{Z}}\cup {\{\stackrel{?}{f}({\lambda }_k)\}}_{k\in \mathbb{Z}}$ along appropriate slowly increasing sequences ${\{t_n\}}_{n\in \mathbb{Z}}$ and ${\{{\lambda }_n\}}_{n\in \mathbb{Z}}$ tending to ±∞ as $n\to \pm \infty$.     

The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.     

A new characterization of Sobolev spaces

Our main result is the following: Let $g\in L^p({\mathbb{R}}^N)$, $1<p<+\infty$, be such that

$\underset{n\in \mathbb{N}}{\sup }\underset{|g(x)?g(y)|>{\delta }_n}{\underset{{\mathbb{R}}^N}{\int }\underset{{\mathbb{R}}^N}{\int }}\frac{{\delta }_n^p}{{|x?y|}^{N+p}}\mathrm{d}x\mathrm{d}y<+\infty ,$

for some arbitrary sequence of positive numbers ${({\delta }_n)}_{n\in \mathbb{N}}$ with ${\lim }_{n\to \infty }{\delta }_n=0$. Then $g\in W^{1,p}({\mathbb{R}}^N)$.This extends a result from H.-M. Nguyen (2006). To cite this article: J. Bourgain, H.-M. Nguyen, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

The nonlinear complexity of level sequences over Z/(4)

For any sequence $\underline{a}$ over $\mathrm{Z}/(2^2)$, there is an unique 2-adic expansion $\underline{a}={\underline{a}}_0+{\underline{a}}_1·2$, where ${\underline{a}}_0$ and ${\underline{a}}_1$ are sequences over $\{0,1\}$ and can be regarded as sequences over the binary field $\mathit{GF}(2)$ naturally. We call ${\underline{a}}_0$ and ${\underline{a}}_1$ the level sequences of $\underline{a}$. Let $f(x)$ be a primitive polynomial of degree $n$ over $\mathrm{Z}/(2^2)$, and $\underline{a}$ be a primitive sequence generated by $f(x)$. In this paper, we discuss how many bits of ${\underline{a}}_1$ can determine uniquely the original primitive sequence $\underline{a}$. This issue is equivalent with one to estimate the whole nonlinear complexity, $\mathit{NL}(f(x),2^2)$, of all level sequences of $f(x)$. We prove that $4n$ is a tight upper bound of $\mathit{NL}(f(x),2^2)$ if $f(x)(\mathrm{mod}2)$ is a primitive trinomial over $\mathit{GF}(2)$. Moreover, the experimental result shows that $\mathit{NL}(f(x),2^2)$ varies around $4n$ if $f(x)(\mathrm{mod}2)$ is a primitive polynomial over $\mathit{GF}(2)$. From this result, we can deduce that $\mathit{NL}(f(x),2^2)$ is much smaller than $L(f(x),2^2)$, where $L(f(x),2^2)$ is the linear complexity of level sequences of $f(x)$.     

On Noether's problem for central extensions of symmetric and alternating groups

For a field k and a finite group G acting regularly on a set of indeterminates $\underset{?}{X}={\{X_g\}}_{g\in G}$, let $k(G)$ denote the invariant field $k{(\underset{?}{X})}^G$. We first prove for the alternating group $A_n$ that, if n is odd, then $\mathbb{Q}(A_n)$ is rational over $\mathbb{Q}(A_{n?1})$. We then obtain an analogous result where $A_n$ is replaced by an arbitrary finite central extension of either $A_n$ or $S_n$, valid over $\mathbb{Q}({\zeta }_N)$ for suitable N. Concrete applications of our results yield: (1) a new proof of Maeda's result on the rationality of $\mathbb{Q}{(X_1,\dots ,X_5)}^{A_5}/\mathbb{Q}$; (2) an affirmative answer to Noether's problem over $\mathbb{Q}$ for both $\stackrel{?}{A_5}$ and $\stackrel{?}{S_5}$; (3) an affirmative answer to Noether's problem over $\mathbb{C}$ for every finite central extension group of either $A_n$ or $S_n$ with $n?5$.