4 problems

Polynomial Stillman-bound conjecture. For all characteristics, the projective dimension of $R/I$ is at worst

Polynomial bound conjecture for ${}^\eta\!A$. For all characteristics, the best possible bound for ${}^\eta\!A$ is, for fixed $\eta$ and fixed $d$, at worst

Polynomial strength-bound conjecture. For all characteristics, the best possible bound for $\mathfrak{K}_d(k)$ is at worst

Let $R$ be a polynomial ring over a field, and let $I$ be an ideal generated by three forms of degree three. Write $\mathop{\mathrm{pd}}\nolimits(R/I)$ for the projective dimension…