[Paper Review] Partial fraction decompositions and an algorithm for computing the vector partition function

Todor Milev|arXiv (Cornell University)|Oct 26, 2009

Advanced Combinatorial Mathematics12 references3 citations

TL;DR

This paper presents two algorithms for computing the vector partition function as a quasipolynomial over pointed polyhedral cones, using partial fraction decompositions in the ring of Laurent polynomials. It provides explicit bounds on the periods of Kostant partition functions for exceptional root systems (E6: 6, E7: 12, E8: 60, F4: 12, G2: 6), and implements the faster algorithm in a publicly available C++ program under the LGPLv3.0 license at vectorpartition.sourceforge.net.

ABSTRACT

This paper gives an exposition of well known results on vector partition functions. The exposition is based on works of M. Brion, A. Szenes and M. Vergne and is geared toward explicit computer realizations. In particular, the paper presents two algorithms for computing the vector partition function with respect to a finite set of vectors $I$ as a quasipolynomial over a finite set of pointed polyhedral cones. We use the developed techniques to relate a result of P. Tumarkin and A. Felikson (and present an independent proof in the particular case of finite-dimensional root systems) to give bounds for the periods of the Kostant partition functions of $E_6$, $E_7$, $E_8$, $F_4$, $G_2$ (the periods are divisors of respectively 6, 12, 60, 12, 6). The first of the described algorithms has been realized and is publicly available under the Library General Public License v3.0 at {http://vectorpartition.sourceforge.net/}. We include (non-unique) partial fraction decompositions for the generating functions of the Kostant partition function for $A_2$, $A_3$, $A_4$, $B_2$, $B_3$, $C_2$, $C_3$, $G_2$ in the appendix.

Motivation & Objective

• To develop explicit, computationally feasible algorithms for computing the vector partition function as a quasipolynomial over polyhedral cones.
• To provide a constructive proof of quasipolynomiality with walls parallel to hyperplanes spanned by rank-n-1 subsets of the input vectors.
• To derive upper bounds on the periods of Kostant partition functions for exceptional Lie algebras using partial fraction decomposition techniques.
• To implement and publicly release a high-performance C++ program for computing vector partition functions and their chamber decompositions.
• To present non-unique partial fraction decompositions for generating functions of Kostant partition functions in classical root systems (A2–A4, B2–B3, C2–C3, G2).

Proposed method

• Uses partial fraction decompositions in the ring ℚ[x₁,…,xₙ, x₁⁻¹,…,xₙ⁻¹] with denominators of the form (1−x^α)^l, avoiding complex roots of unity for computational efficiency.
• Employs the elongated Szenes-Vergne formula (Lemma 2.1(c)) to generate initial partial fraction decompositions.
• Applies a reduction algorithm (Section 6.1.1) that iteratively simplifies fractions using the identity 1/(1−x^α) = (1+x^α)/(1−x^{2α}) to eliminate short root denominators.
• Utilizes a two-step algorithm: first, a recursive reduction process to simplify rational functions; second, a transformation to eliminate redundant short roots.
• Employs a custom C++ implementation with template classes for polynomials, rational arithmetic, hash tables, and polyhedral cone subdivisions.
• Validates correctness via checksums: substituting rational values before and after each transformation to detect programming errors.

Experimental results

Research questions

• RQ1 What is the structure of the vector partition function as a quasipolynomial over a finite set of pointed polyhedral cones?
• RQ2 How can partial fraction decompositions be systematically computed for generating functions of vector partition functions without complex roots of unity?
• RQ3 What are the exact periods of the Kostant partition functions for the exceptional root systems E6, E7, E8, F4, and G2?
• RQ4 Can an efficient, publicly available algorithm be implemented to compute vector partition functions and their chamber decompositions?
• RQ5 How do the two proposed algorithms compare in performance, and what modifications improve efficiency for classical root systems?

Key findings

• The vector partition function P_I(γ) is a quasipolynomial over a finite set of closed, pointed polyhedral cones with walls parallel to hyperplanes spanned by rank-n-1 subsets of I.
• The algorithm in Sections 2 and 3 (based on partial fraction decomposition) is significantly faster in practice than the elementary algorithm in Section 4.
• The periods of the Kostant partition functions for E6, E7, E8, F4, and G2 are bounded above by 6, 12, 60, 12, and 6 respectively, and are divisors of these values.
• The modified algorithm in Section 6.1.1, tailored for classical root systems, outperforms the general algorithm in Section 2.3 due to optimized reduction rules.
• The implementation of the vector partition function program is publicly available at http://vectorpartition.sourceforge.net/ under the LGPLv3.0 license.
• Non-unique partial fraction decompositions for the generating functions of Kostant partition functions are explicitly computed and listed in the appendix for A2, A3, A4, B2, B3, C2, C3, and G2.