Genuine and strongly genuine polynomials: With an application to the persistence of Galois groups under specialization

We develop the theory of strongly $n$-genuine polynomials $F(Y,X_1,\dots ,X_n)$, which have the property that the number of specializations $F(Y,X_1,{\mathbf{x}}^{\mathrm{\prime }})$ with ${\mathbf{x}}^{\mathrm{\prime }}=(x_2,\dots ,x_n)\in {\mathbb{Z}}^{n-1}$ (respectively ${\mathbf{x}}^{\mathrm{\prime }}\in {\mathbb{F}}_p^{n-1}$) such that $F(Y,X_1,{\mathbf{x}}^{\mathrm{\prime }})$ is reducible over $\bar{\mathbb{Q}}$ (respectively over ${\bar{\mathbb{F}}}_p$) can be well-controlled quantitatively. We also develop the theory of a larger class of $n$-genuine polynomials $F(Y,X_1,\dots ,X_n)$, which have the property that the number of specializations $F(Y,X_1,{\mathbf{x}}^{\mathrm{\prime }})$ with ${\mathbf{x}}^{\mathrm{\prime }}\in {\mathbb{Z}}^{n-1}$ (respectively ${\mathbf{x}}^{\mathrm{\prime }}\in {\mathbb{F}}_p^{n-1}$) such that $F(Y,X_1,{\mathbf{x}}^{\mathrm{\prime }})$ splits completely over $\bar{\mathbb{Q}}$ (respectively over ${\bar{\mathbb{F}}}_p$) into factors that are linear in $Y$ can be well-controlled quantitatively. For each of these classes, we prove that there are four equivalent characterizations. As an application, we demonstrate that $n$-genuine and strongly $n$-genuine polynomials can be used to prove, for any polynomial $F(Y,X_1,\dots ,X_n)$, an upper bound for the number of specializations $F(Y,\mathbf{x})$ with $\mathbf{x}=(x_1,\dots ,x_n)\in {\mathbb{Z}}^n$ such that the Galois group of the splitting field of $F(Y,\mathbf{x})$ over $\mathbb{Q}$ is not isomorphic to the Galois group of the splitting field of $F(Y,X_1,\dots ,X_n)$ over $\mathbb{Q}(X_1,\dots ,X_n)$. We simultaneously prove analogous results over any number field.