9 problems

Let $p$ be an odd prime and let … which is a singular ring of characteristic $p$ and dimension $d-1$. For a Noetherian local ring $(R,\mathfrak{m})$ of characteristic $p$, write…

Use the notation above: let $d\geq1$, let $p>2$ be prime, define … and let $(A,\mathfrak m,k)$ be a $d$-dimensional unmixed local ring with $k=\overline{\mathbb F_p}$. Higher-dimen…

Let $d\geq1$ be an integer and let $p>2$ be prime. Define … Let $(A,\mathfrak m,k)$ be a $d$-dimensional unmixed local ring with $k=\overline{\mathbb F_p}$. Han–Monsky lower-bound…

Veronese lower-bound conjecture. Then

Let $d\geq 2$. Set … where … Let $(R,\mathfrak{m},k)$ be an unmixed non-regular local ring of characteristic $p$ and dimension $d$. Here, unmixed means that…

Let $(R,\mathfrak{m},K)$ be a local ring. Let $I\subseteq R$ be an ideal generated by a part of a system of parameters $(f_1,\ldots,f_\ell)$, and set $\overline{R}=R/I$. Let $J$ be…

Let $k$ be a field of characteristic $p\geq 3$, let $d\geq 2$, and set … For an arbitrary $d$-dimensional unmixed local ring $(A,\mathfrak{m})$ of characteristic $p$ with residue f…

Let $A=\mathbb{F}_2[[U,V,X,Y,Z]]/(UV+X^3+Y^3+XYZ)$. Monsky's conjecture. The Hilbert–Kunz multiplicity of $A$ is … This is a specific predicted value for a notable Hilbert–Kunz mul…

Watanabe–Yoshida conjecture. If $R$ is not regular, then