19 problems

For fixed $d\geq 1$ and each prime $p>2$, let … and let $\operatorname{e_{HK}}(A_{p,d})$ be its Hilbert–Kunz multiplicity. Yoshida's conjecture. The sequence…

For $d\geq2$, let … be the degree-$d$ Fermat hypersurface ring, and define $c_n$ and $c'_n$ by … … Here $\xi_{2d}$ is a primitive $2d$-th root of unity. Rationality conjecture. The…

Smirnov's conjecture. Assume further that $R$ is analytically unramified and Cohen–Macaulay. Then

Let $(R,\mathfrak m)$ be a noetherian local ring of prime characteristic. Let $f_1,f_2,\ldots,f_r$ be part of a system of parameters, let $J$ be an ideal generated by a full system…

Let $(R,\mathfrak m)$ be a noetherian local ring of prime characteristic. Let $f_1,f_2,\ldots,f_r$ be part of a system of parameters, let $J$ be an $\mathfrak m$-primary ideal, and…

Let $(R,\mathfrak m)$ be a Cohen–Macaulay local ring of dimension $d$, and let $I$ be an $\mathfrak m$-primary ideal. The ideal $I$ is stable if it satisfies $I^2=QI$ for a minimal…

For each fixed integer $n$, let $R_{p,n+1}$ be the quadric hypersurface ring of dimension $n+1$ in characteristic $p$, and let ${\bf m}$ be its homogeneous maximal ideal. Yoshida's…

Let $(A,\mathfrak{m})$ be a non-regular local ring of dimension $d \geq 1$, and let $A_{p,d}$ be the simple singularity defined above. Maximal F-signature conjecture. … Since…

Let $(A,\mathfrak{m}, k)$ be a formally unmixed non-regular local ring of dimension $d \geq 1$ with algebraically closed residue field. Let $m \geq 1$ be an integer, and let…

Let $(R,\mathfrak m)$ be an F-rational local ring. Write $\operatorname{s_{rel}}(R)$ for its relative F-rational signature and $\operatorname{s_{dual}}(R)$ for its dual F-rational…

Hilbert–Kunz strengthening.

Let $(R, \mathfrak m)$ be a Cohen–Macaulay local ring of dimension $d$ and let $I$ be an $\mathfrak m$-primary ideal. Write $I^{[q]}$ for the Frobenius power of $I$, and let…

Throughout, let $R:=\bigoplus_{n\geq 0}R_n$ be a standard graded algebra over a field of prime characteristic $p>0$, let $\mathfrak{m}:=\bigoplus_{n>0}R_n$ be its irrelevant ideal,…

Upper semi-continuity conjecture. If $R$ is an excellent ring, then the function

Generic Hilbert–Kunz multiplicity conjecture. The generic Hilbert–Kunz multiplicity of a homogeneous hypersurface of degree $delta$ and ring dimension $t$ is

Transcendental Hilbert–Kunz multiplicity conjecture. There exist examples of this type for which the Hilbert–Kunz multiplicity is transcendental and the second coefficient in the H…

Vanishing conjecture. If $i\le d/2$, then $\beta_i=0$.

The $\phi_f$ conjecture. The element

Let $f=x^{3}+y^{3}+xyz$ be the element of $(\mathbb{Z}/2)[x,y,z]$ defining a nodal cubic. Let $\phi_f$ denote the associated function, and let $\phi_0$ be the function defined recu…