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A Conjecture In the following, a polynomial $P(x)$ is written in Bernstein form of degree $n$ if it is written as— $$P(x)=\sum_{k=0}^n a_k {n \choose k} x^k (1-x)^{n-k},$$ where $a_0, ..., a_n$ are ...

Let $f\in\operatorname{int}(\mathbb Z)$, the ring of integer-valued rational polynomials. Define $\operatorname{P}^+(f)$ as the number of primes $>0$ that $f$ assumes at distinct integral arguments....

For simplicity this is about polynomials in just two variables. Any $f\in\mathbb Q[X,Y]$ can be written as a linear combination of monomials $X^iY^j$ and therefore as a sum of polynomials $p_{ij}\...

Littlewood in [L] states several conjectures regarding asymptotics of polynomials with $\pm1$ coefficients. He considers the class $\mathscr F$ of polynomials of form $\sum^n\pm z^m$ and asks whether ...

Let $T:\mathbb Q(x)\to \mathbb Q(x)$ be the operator of inverse logarithmic derivative, i.e. $$Tf=\frac{f}{f'}.$$ Define $$p_n(x)=T^n\left(x-\frac{x^2}{2}\right).$$ Let $f_n(x) \in \mathbb Z[x]$ be ...