11 problems

A function $f$ over finite fields of characteristic $2$ is exceptional APN if it is APN over infinitely many extensions of the base field. The Gold functions are $f(x)=x^{2^k+1}$,…

Conjecture. For general $N$, a ratio vector determines unique real numbers $r_{3},\ldots,r_{N}$ such that the polynomial $p$ has that ratio vector.

Let $\mathbb{F}_{q}=\mathbb{F}_{3^m}$, with $m$ odd, and let $f(x)=x^d$ be a monomial over $\mathbb{F}_{q}$. Consider the two families of exponents … and … For a polynomial $f$ ove…

Monomial intersection-distribution conjecture. The following two families of monomials $x^d$ over $\mathbb{F}_q=\mathbb{F}_{3^m}$ have the same intersection distribution as $x^3$:

Gács–Nagy–Hegedűs–Pálvölgyi conjecture. If there is no polynomial with range $M$ of degree less than $q-m$, then $M$ contains an element of multiplicity at least $q-m$.

Bijectivity conjecture. This difficulty is intimately connected with the absence of a bijective property.

Let ${\mathbf{A}}$ and ${\mathbf{B}}$ be expanded groups such that every congruence of ${\mathbf{A}}\times{\mathbf{B}}$ is a product congruence. Let $f$ and $g$ be unary polynomial…

Let $q$ be a prime power, let $cmathbb F_q$ be the finite field of cardinality $q$, and let $G(q)$ be the subgroup of permutations of $cmathbb F_q^*$ generated by the permutations…

Gács et al.'s conjecture. Let $k<\sqrt{p}$. If no polynomial with range $M$ has degree less than $q-k$, then $M$ contains an element of multiplicity at least $q-k$.

For odd $d\ge5$, let $Q_d(x)$ be the function discussed in the preceding numerical analysis, and let $x_{\max}(d)$ and $x_{\min}(d)$ be real numbers. Monotonicity and asymptotic ex…

Let $m,n\in\mathbb{N}$, and consider functions from $\mathbb{Z}_m^n$ to $\mathbb{Z}_m$. Newton interpolation is the interpolation algorithm in which successive divisions are perfor…