[Paper Review] The parametric Frobenius problem and parametric exclusion

Bobby Shen|arXiv (Cornell University)|Oct 5, 2015

Commutative Algebra and Its Applications5 references3 citations

TL;DR

This paper establishes that the parametric Frobenius problem—generalizing the classical Frobenius number to polynomial sequences—yields functions that are eventually quasi-polynomial (EQP) in the parameter t. By introducing a novel framework of parametric exclusion problems and leveraging parametric integer linear programming, the authors prove that both the ℓ-th largest integer with fewer than m representations and the count of such integers are EQP functions of t, resolving a conjecture by Roune and Woods for fixed m and ℓ.

ABSTRACT

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 $\ell^{ ext{th}}$ 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 $\ell^{ ext{th}}$ 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 $t.$

Motivation & Objective

• To generalize the classical Frobenius problem to polynomial sequences in a parameter t.
• To resolve a conjecture by Roune and Woods on the eventual quasi-polynomial nature of the parametric Frobenius number.
• To develop a framework—parametric exclusion problems—that unifies and generalizes parametric integer programming for representation-counting problems.
• To establish that both the ℓ-th largest integer with fewer than m representations and the count of such integers are eventually quasi-polynomial functions of t.

Proposed method

• Introduce the parametric exclusion problem as a generalization of the parametric Frobenius problem.
• Define f_{m,ℓ}(P₁(t),…,Pₙ(t)) as the ℓ-th largest multiple of gcd(P₁(t),…,Pₙ(t)) with fewer than m nonnegative integer representations.
• Define g_m(P₁(t),…,Pₙ(t)) as the number of positive multiples of gcd(P₁(t),…,Pₙ(t)) with fewer than m representations.
• Prove that f_{m,ℓ}(P₁(t),…,Pₙ(t)) and g_m(P₁(t),…,Pₙ(t)) are eventually quasi-polynomial functions of t using a decomposition into parametric integer linear programs.
• Use the structure of the solution set of the parametric integer program to show that the functions depend on t modulo a fixed period d, with components given by polynomials R₀,…,R_{d−1}.
• Leverage the fact that the number of representations h(k) grows without bound as k increases, ensuring the finiteness of the sets being analyzed.

Experimental results

Research questions

• RQ1 Is the ℓ-th largest integer with fewer than m nonnegative integer representations of the form ∑b_i P_i(t) an eventually quasi-polynomial function of t for fixed m and ℓ?
• RQ2 Does the count of positive integers with fewer than m such representations also form an eventually quasi-polynomial function of t?
• RQ3 Can the parametric Frobenius problem be systematically reduced to a parametric integer linear programming problem with structured solution sets?
• RQ4 What is the relationship between the components of the resulting eventually quasi-polynomial functions and the underlying algebraic structure of the generating polynomials?
• RQ5 Are there effective bounds on the period of the eventual quasi-polynomial behavior or algorithmic methods to compute the components?

Key findings

• The function f_{m,ℓ}(P₁(t),…,Pₙ(t)), which gives the ℓ-th largest integer with fewer than m representations as a nonnegative integer combination of P₁(t),…,Pₙ(t), is eventually quasi-polynomial in t.
• The function g_m(P₁(t),…,Pₙ(t)), counting the number of positive multiples of gcd(P₁(t),…,Pₙ(t)) with fewer than m representations, is also eventually quasi-polynomial in t.
• The result confirms a conjecture by Roune and Woods for fixed m and ℓ, showing that F(P₁(t),…,Pₙ(t)) is eventually quasi-polynomial when the P_i(t) are integer-valued polynomials with positive leading coefficients.
• The proof relies on transforming the parametric Frobenius problem into a parametric exclusion problem, which is then analyzed via parametric integer linear programming.
• The components of the eventual quasi-polynomial are determined by the structure of the solution set of the parametric system, with periodic behavior modulo a fixed d.
• The framework shows that even when m and ℓ are replaced by polynomials in t, the resulting functions are not necessarily eventually quasi-polynomial, as demonstrated by counterexamples.