Add to ORKGGiven two polynomials $P(\underline x)$, $Q(\underline x)$ in one or more variables and with integer coefficients, how does the property that they are coprime relate to their values $P(\underline n), Q(\underline n)$ at integer points $\underline n$ being coprime? We show that the set of all $\gcd (P(\underline n), Q(\underline n))$ is stable under gcd and under lcm. A notable consequence is a result of Schinzel: if in addition $P$ and $Q$ have no fixed prime divisor (i.e., no prime dividing all values $P(\underline n)$, $Q(\underline n)$), then $P$ and $Q$ assume coprime values at "many" integer points. Conversely we show that if "sufficiently many" integer points yield values that are coprime (or of small gcd) then the original polynomial...
summary:Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$...
Let ∈ℤ[] be a quadratic or cubic polynomial. We prove that there exists an integer ⩾2 such that ...
International audienceAbstract We establish a version “over the ring” of the celebrated Hilbert Irre...
Flip a coin to select a random polynomial over F2. The sequence HTHHHTH, for example, corresponds to...
If $\mathcal{A}\subset\mathbb{N}$ is such that it does not contain a subset $S$ consisting of $k$ pa...
International audienceIf $\mathcal{A}\subset\mathbb{N}$ is such that it does not contain a subset $S...
The concept of divisibility naturally led to the concepts of primality, common divisors and polynomi...
AbstractWe consider the following computational problem: we are given two coprime univariate polynom...
AbstractFrom the work of S. Corteel et al. (1998, J. Combin. Theory Ser. A82, 186–192), the number o...
Abstract. Univariate polynomials with only real roots – while special – do occur often enough that t...
Abstract. Univariate polynomials with only real roots – while special – do occur often enough that t...
The set P(n) of all primes equal to or less than n has the obvious property that it contains exactly...
In [JM90] Jankowski and Marlewski prove by elementary methods that if f and g are polynomials in Q[X...
A well-known result of Ehrenfeucht states that a difference polynomial f(X)-g(Y) in two variables X,...
summary:Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$...
summary:Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$...
Let ∈ℤ[] be a quadratic or cubic polynomial. We prove that there exists an integer ⩾2 such that ...
International audienceAbstract We establish a version “over the ring” of the celebrated Hilbert Irre...
Flip a coin to select a random polynomial over F2. The sequence HTHHHTH, for example, corresponds to...
If $\mathcal{A}\subset\mathbb{N}$ is such that it does not contain a subset $S$ consisting of $k$ pa...
International audienceIf $\mathcal{A}\subset\mathbb{N}$ is such that it does not contain a subset $S...
The concept of divisibility naturally led to the concepts of primality, common divisors and polynomi...
AbstractWe consider the following computational problem: we are given two coprime univariate polynom...
AbstractFrom the work of S. Corteel et al. (1998, J. Combin. Theory Ser. A82, 186–192), the number o...
Abstract. Univariate polynomials with only real roots – while special – do occur often enough that t...
Abstract. Univariate polynomials with only real roots – while special – do occur often enough that t...
The set P(n) of all primes equal to or less than n has the obvious property that it contains exactly...
In [JM90] Jankowski and Marlewski prove by elementary methods that if f and g are polynomials in Q[X...
A well-known result of Ehrenfeucht states that a difference polynomial f(X)-g(Y) in two variables X,...
summary:Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$...
summary:Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$...
Let ∈ℤ[] be a quadratic or cubic polynomial. We prove that there exists an integer ⩾2 such that ...
International audienceAbstract We establish a version “over the ring” of the celebrated Hilbert Irre...