Related papers: Polynomials with maximal differential uniformity a…

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0 \in \F_2 [x]$. If $f_0$ is of degree $n = 2^l \cdot m$, where $m$ is odd and $l$ is a non-negative integer,…

A-polynomials were introduced by Meyn and play an important role in the iterative construction of high degree self-reciprocal irreducible polynomials over the field F_2, since they constitute the starting point of the iteration. The exact…

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta\sqrt{n} \leqslant |P(z)| \leqslant…

Let $f\in\mathbb{Z}[X]$ be quadratic or cubic polynomial. We prove that there exists an integer $G_f\geq 2$ such that for every integer $k\geq G_f$ one can find infinitely many integers $n\geq 0$ with the property that none of…

A key property of an algebraic variety is whether it is absolutely irreducible, meaning that it remains irreducible over the algebraic closure of its defining field, and determining absolute irreducibility is important in algebraic geometry…

We prove that with high probability, d+1 random Bernoulli polynomials in d variables of degree n (n goes to infinity) do not possess a common root.

A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients.…

A classic result by Carbery and Wright states that a polynomial of Gaussian random variables exhibits anti-concentration in the following sense: for any degree $d$ polynomial $f$, one has the estimate $P( |f(x)| \leq \varepsilon \cdot…

Let $A = a_0T^m + \sum_{j=1}^{m-1} a_j (T^{m-j}+T^{m+j}) + T^{2m}+1 \in \mathbf{Z}[T]$ be a monic reciprocal polynomial of degree $2m$ sampled randomly by selecting its coefficients $a_0,a_1,\dots,a_{m-1}$ independently according to a given…

We study a conjecture called "linear rank conjecture" recently raised in (Tsang et al., FOCS'13), which asserts that if many linear constraints are required to lower the degree of a GF(2) polynomial, then the Fourier sparsity (i.e. number…

We define the second discriminant $D_2$ of a univariate polynomial $f$ of degree greater than $2$ as the product of the linear forms $2\,r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$ with $i<j$ and $j\neq k, k\neq i$. $D_2$…

We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d.…

We show that $0,1$-polynomials of high degree and few terms are irreducible with high probability. Formally, let $k\in\mathbb{N}$ and $F(x)=1+\sum_{i=1}^kx^{n_i}$, where $ 0<n_1<\cdots<n_k\leq N. $ Then we show that…

When monic integral polynomials of degree $n \geq 2$ are ordered by the maximum of the absolute value of their coefficients, the Hilbert irreducibility theorem implies that asymptotically 100% are irreducible and have Galois group…

We consider absolutely irreducible polynomials $f \in Z[x,y]$ with $\deg_x(f)=m$, $\deg_y(f)=n$ and height $H$. We show that for any prime $p$ with $p>c_{mn} H^{2mn+n-1}$ the reduction $f \bmod p$ is also absolutely irreducible. Furthermore…

Assume that the Riemann hypothesis holds for Dedekind zeta functions. Under this assumption, we prove that a degree $d$ polynomial with random multiplicative $\pm1$ coefficients is irreducible in $\mathbb{Z}[x]$ with probability…

We give all the polynomials functions of degree 20 which are APN over an infinity of field extensions and show they are all CCZ-equivalent to the function $x^5$, which is a new step in proving the conjecture of Aubry, McGuire and Rodier.

In this paper, by the Hasse-Weil bound, we determine the necessary and sufficient condition on coefficients $a_1,a_2,a_3\in\mathbb{F}_{2^n}$ with $n=2m$ such that $f(x) = {x}^{3\cdot2^m} + a_1x^{2^{m+1}+1} + a_2 x^{2^m+2} + a_3x^3$ is an…

We give all bi-unitary non splitting even perfect polynomials over the prime field of two elements, which are divisible by Mersenne irreducible polynomials raised to special exponents. We also identify all bi-unitary perfect polynomials…

We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for…