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The parametric Frobenius problem and parametric exclusion

The Frobenius number of relatively prime positive integers $a_1, \ldots, a_n$ is the largest integer that is not a nononegative integer combination of the $a_i.$ Given positive integers $a_1, \ldots, a_n$ with $n \ge 2,$ the set of multiples of $\gcd(a_1, \ldots, a_n)$ which have less than $m$ distinct representations as a nonnegative integer combination of the $a_i$ is bounded above, so we define $f_{m, \ell}(a_1, \ldots, a_n)$ to be the $\ell^{\text{th}}$ largest multiple of $\gcd(a_1, \ldots, a_n)$ with less than $m$ distinct representations (which generalizes the Frobenius number) and $g_m(a_1, \ldots, a_n)$ to be the number of positive multiples of $\gcd(a_1, \ldots, a_n)$ with less than $m$ distinct representations. In the parametric Frobenius problem, the arguments are polynomials. Let $P_1, \ldots, P_n$ be integer valued polynomials of one variable which are eventually positive. We prove that $f_{m, \ell}(P_1(t), \ldots, P_n(t))$ and $g_m(P_1(t), \ldots, P_n(t)),$ as functions of $t,$ are eventually quasi-polynomial. A function $h$ is eventually quasi-polynomial if there exist $d$ and polynomials $R_0, \ldots, R_{d-1}$ such that for such that for sufficiently large integers $t,$ $h(t)=R_{t \pmod{d}}(t).$ We do so by formulating a type of parametric problem that generalizes the parametric Frobenius Problem, which we call a parametric exclusion problem. We prove that the $\ell^{\text{th}}$ largest value of some polynomial objective function, with multiplicity, for a parametric exclusion problem and the size of its feasible set are eventually quasi-polynomial functions of $t.$