It is shown that for any Artinian modules $M$, $dim{M}^{\vee }$ is the greatest integer $i$ such that ${\text{H}}_{𝔪}^i\left(M^{\vee }\right)\ne 0$.

Let $X$ be a projective Frobenius split variety with a fixed Frobenius splitting $\theta$. In this paper we give a sharp uniform bound on the number of subvarieties of $X$ which are compatibly Frobenius split with $\theta$. Similarly, we give a bound on the number of prime $F$-ideals of an $F$-finite $F$-pure local ring. Finally, we also give a bound on the number of log canonical centers of a log canonical pair. This final variant extends a special case of a result of Helmke.

We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if $W$ is such a variety, then every piecewise polynomial function on $W$ can be written as suprema of infima of polynomial functions on $W$. More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of such varieties, which implies the Pierce-Birkhoff Conjecture.

Given integers $D\>0,n\>1,0\<r\<n$ and a constant $C\>0$, consider the space of $r$-tuples $\overrightarrow{P}=(P_1...P_r)$ of real polynomials in $n$ variables of degree $\le D$, whose coefficients are $\le C$ in absolute value, and satisfying $\det {\left(\frac{\partial P_i}{\partial x_i}\left(0\right)\right)}_{1\le i,j\le r}=1$. We study the family $\left\{f\right|V\}$ of algebraic functions, where $f$ is a polynomial, and $V=\left\{\right|x|\le \delta ,\overrightarrow{P}(x)=0\},\delta \>0$ being a constant depending only on $n,D,C$. The main result is a quantitative extension theorem for these functions which is uniform in $\overrightarrow{P}$. This is used to prove Bernstein-type inequalities which are again uniform with respect to $\overrightarrow{P}$.The proof is based on...

When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$, with homogeneous maximal ideal $𝔪$, it is known that for an ideal $I$ of $S$, the regularity of powers of $I$ becomes eventually a linear function, i.e., $\mathrm{reg}\left(I^m\right)=dm+e$ for $m\gg 0$ and some integers $d$, $e$. This motivates writing $\mathrm{reg}\left(I^m\right)=dm+e_m$ for every $m\ge 0$. The sequence $e_m$, called the defect sequence of the ideal $I$, is the subject of much research and its nature is still widely unexplored. We know that $e_m$ is eventually constant. In this article, after...