7 problems

Let a semi-regular sequence mean a sequence of homogeneous polynomials satisfying the semi-regularity condition. Pardue's conjecture. Semi-regular sequences are sequences of generi…

Let $m$ be a positive integer and consider sequences of $m$ polynomials in $\mathbb{F}_2[x_1,\ldots,x_n]$. The term semi-regular refers to the semi-regularity condition for polynom…

Let $m$ be a positive integer and let $d_1,\ldots,d_m$ be the degrees of homogeneous polynomials in $\mathbb{F}_2[x_1,\ldots,x_n]$. A sequence is cryptographic semi-regular if it s…

Fix an integer $d>1$. For integers $n$ and $m$, let $\pi(n,m,d)$ be the proportion of sequences of degree $d$ and length $m$ in $n$ variables that are semi-regular. Fixed-degree ge…

For integers $n$ and $d$ with $1\leq d\leq n$, let $\pi(n,d)$ be the proportion of sequences of degree $d$ and length $n$ in $n$ variables that are semi-regular. Degree-d genericit…

For integers $n,m,d_1,\dots,d_m$, let $E(n,m,d_1,\dots,d_m)$ be the set of algebraic systems of $m$ equations of degrees $d_1,\dots,d_m$ in $n$ variables over ${\mathbb F}_2$, and…

Let a semi-regular sequence be a sequence of homogeneous elements in the relevant polynomial-ring setting, and let the number of variables tend to infinity. Bardet–Faugère–Salvy–Ye…