Friability of the values of a polynomial
For an integer \(n\neq 0\) let \(P^+(n)\) denote the greatest prime factor of \(n\) if \(n\neq \pm 1\), \(P^+(n)=1\) if \(n=\pm 1\) . Let \(y\geq 1\). A non-zero integer \(n\) is called \(y\)-friable if \(P^+(n)\leq y\). For \(d>1,n>1\) integers, let \[ \Delta(d,n):=\sup\{\delta\in{\mathbb R}\mid \forall F\in{\mathbb Z}[X_1,\ldots,X_n]\,\text{absolutely irreducible}, \] \[ \deg F=d, \exists x_0=x_0(F), \exists c_0=c_0(F)>0: \Psi_F(x,x^{d-\delta})\geq c_0x^n,\forall x>x_0\}, \] where for positive reals \(x,y\) \[ \Psi_F(x,y):=\#\{(m_1,\ldots,m_n)\in{\mathbb Z}^n:\max_{1\leq i\leq n}|m_i|\leq x, \] \[ F(m_1,\ldots,m_n)\neq 0\,\text{and}\,P^+(F(m_1,\ldots,m_n))\leq y \}. \] The author proves that \[ \Delta(d,n)\geq\frac43 \] for \(d\geq 3\),\(n\geq2\). He improves the bound under some geometrical assumptions on the polynomial \(F\).
values of polynomials in several variables
scientific article; zbMATH DE number 5822868