The reviewed record of science sign in
Pith

arxiv: 2605.05319 · v1 · submitted 2026-05-06 · math.CO

Induced Lorentzian and volume polynomials

Christopher Eur , Nutan Nepal , Daniel Qin This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-05-08 16:09 UTCgrok-4.3open to challenge →

classification math.CO
keywords Lorentzian polynomialsvolume polynomialsinduced polymatroidspanel countingcombinatorial enumerationlog-concavitypolynomial preservers
0
0 comments X

The pith

The numbers of valid expert panels for each topic subset form a Lorentzian polynomial.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

For a group of people with expertise covering various topics, the counts of panels that can cover each possible subset of topics form the coefficients of a Lorentzian polynomial. The proof introduces an inducing operator on polynomials, linked to induced polymatroids, and demonstrates that this operator preserves both Lorentzian polynomials and realizable volume polynomials. A reader would care because Lorentzian polynomials come with useful properties like log-concavity of coefficients, leading to inequalities that the panel counts must satisfy. This approach generates new examples of such polynomials from familiar ones without requiring direct geometric constructions.

Core claim

Suppose one has a party of m people, whose expertise collectively covers n topics. Given a subset T of the topics, one wishes to form a panel of |T| people from the party such that T can be covered by assigning a distinct topic to each panel member with the expertise. We show that the numbers of such panels, as T varies, form a Lorentzian polynomial. We achieve this by showing that a certain linear operator on polynomials, which we call the inducing operator for its connection to induced (poly)matroids, preserves Lorentzian polynomials and realizable volume polynomials.

What carries the argument

The inducing operator, a linear operator on polynomials connected to induced polymatroids, that preserves the class of Lorentzian polynomials and realizable volume polynomials.

If this is right

  • Panel counting numbers obey log-concavity and other Lorentzian inequalities.
  • Applying the inducing operator to any realizable volume polynomial yields a Lorentzian polynomial.
  • Induced structures from polymatroids produce Lorentzian polynomials systematically.
  • The method applies to counting problems in induced matroids and related combinatorial objects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This operator might be used to prove Lorentzian properties for other counting polynomials in matroid theory.
  • Computational verification on small expertise graphs could provide evidence for the preservation result.
  • Links to applications in fair division or assignment problems where panel formation is relevant.

Load-bearing premise

The panel-counting polynomial is obtained by applying the inducing operator to a Lorentzian polynomial or a realizable volume polynomial coming from the induced polymatroid structure of the expertise relation.

What would settle it

Compute the panel-counting polynomial for a small specific instance of people and topics where the expertise does not yield a Lorentzian polynomial, such as by violating the quadratic log-concavity condition on the coefficients.

read the original abstract

Suppose one has a party of $m$ people, whose expertise collectively covers $n$ topics. Given a subset $T$ of the topics, one wishes to form a panel of $|T|$ people from the party such that $T$ can be covered by assigning a distinct topic to each panel member with the expertise. We show that the numbers of such panels, as $T$ varies, form a Lorentzian polynomial. We achieve this by showing that a certain linear operator on polynomials, which we call the ``inducing operator'' for its connection to induced (poly)matroids, preserves Lorentzian polynomials and realizable volume polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces an 'inducing operator' on multivariate polynomials, motivated by induced polymatroids from bipartite expertise graphs (people to topics). It proves that this linear operator preserves the class of Lorentzian polynomials and the class of realizable volume polynomials. The central application is that the polynomial whose coefficients count, for each subset T of topics, the number of panels of size |T| that cover T via distinct expert assignments, is Lorentzian.

Significance. If the preservation result holds, the work enlarges the supply of explicitly constructible Lorentzian polynomials and supplies a new combinatorial source (panel-counting polynomials) for them. The connection to induced polymatroids and the explicit linear operator may be reusable for other counting problems in matroid theory and algebraic combinatorics. The manuscript supplies a concrete, falsifiable prediction: the panel polynomial satisfies the Lorentzian inequalities.

major comments (2)
  1. [§3] §3, Definition 3.2 and Theorem 3.4: the inducing operator is defined via deletion/contraction on the polymatroid lattice induced by the bipartite incidence matrix; however, the manuscript does not verify that the base polynomial to which the operator is applied is the full matroid polynomial rather than a truncation or restriction. A mismatch here would mean the panel-counting polynomial is not the image under the operator, so the transfer of the Lorentzian property fails.
  2. [§4] §4, Proposition 4.1: the claim that the expertise relation produces an induced polymatroid whose generating function, after the inducing operator, exactly equals the multivariate panel-counting polynomial, is asserted without an explicit bijection or generating-function identity relating the two sides. This identification is load-bearing for the main theorem.
minor comments (2)
  1. [§2] The notation for the inducing operator (e.g., the symbol and the precise domain) is introduced in §2 but used without reminder in later sections; a short recap table would improve readability.
  2. Several citations to the Lorentzian-polynomial literature (Brändén-Huh, etc.) are given only by author-year; full bibliographic details should be supplied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The two major points raised concern the precise identification of the base polynomial and the explicit combinatorial correspondence in the application to panel-counting polynomials. We address each below and will incorporate clarifications into the revised version.

read point-by-point responses

  1. Referee: [§3] §3, Definition 3.2 and Theorem 3.4: the inducing operator is defined via deletion/contraction on the polymatroid lattice induced by the bipartite incidence matrix; however, the manuscript does not verify that the base polynomial to which the operator is applied is the full matroid polynomial rather than a truncation or restriction. A mismatch here would mean the panel-counting polynomial is not the image under the operator, so the transfer of the Lorentzian property fails.

    Authors: We appreciate the referee drawing attention to this point of precision. The inducing operator in Definition 3.2 is constructed directly from the rank function of the full induced polymatroid on the bipartite incidence structure; the deletion and contraction operations are applied to the complete lattice without any preliminary truncation or restriction. Consequently, the base polynomial to which the operator is applied is the generating polynomial of this full polymatroid. To eliminate any possible ambiguity, we will insert a brief clarifying lemma immediately after Definition 3.2 that explicitly confirms the base is the full polynomial by relating its coefficients to the rank function of the induced polymatroid. This addition will ensure that the image under the operator is precisely the panel-counting polynomial, so the Lorentzian property transfers as stated in Theorem 3.4. revision: yes

  2. Referee: [§4] §4, Proposition 4.1: the claim that the expertise relation produces an induced polymatroid whose generating function, after the inducing operator, exactly equals the multivariate panel-counting polynomial, is asserted without an explicit bijection or generating-function identity relating the two sides. This identification is load-bearing for the main theorem.

    Authors: We agree that an explicit combinatorial identification strengthens the argument. The equality in Proposition 4.1 follows from the fact that each valid panel assignment for a topic subset T corresponds to a basis of the induced polymatroid, with the multivariate generating function obtained by applying the inducing operator to the polymatroid polynomial. In the revised manuscript we will expand the proof of Proposition 4.1 to include (i) a direct bijection between the counted panels and the bases of the induced structure and (ii) the explicit generating-function identity obtained by summing the operator’s action over the relevant monomials. This expanded proof will be self-contained and will not alter the statement or the subsequent application of the preservation theorems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new operator and preservation proof are independent of target counts

full rationale

The paper introduces the inducing operator as a new linear operator on polynomials, proves that it preserves the Lorentzian property and realizable volume polynomials, and then applies it to the generating function of the induced polymatroid arising from the bipartite expertise relation. The target panel-counting polynomial is defined as the image under this operator. No equation or step equates the final coefficients to the inputs by construction, renames a fitted quantity, or reduces the Lorentzian claim to a self-citation chain. The base polynomial's Lorentzian status is taken from external prior literature on matroids/polymatroids rather than derived within the paper. This is the normal case of a self-contained derivation transferring a known property via a newly proven operator.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the existence of the inducing operator and its preservation properties for the class of Lorentzian polynomials and realizable volume polynomials; these are introduced in the paper rather than derived from first principles.

axioms (2)
  • domain assumption Lorentzian polynomials are closed under the inducing operator when the operator arises from induced polymatroid structures.
    Invoked to conclude that the panel-counting polynomial is Lorentzian.
  • domain assumption Realizable volume polynomials are preserved by the same operator.
    Used to extend the result beyond pure Lorentzian case.
invented entities (1)
  • Inducing operator no independent evidence
    purpose: Linear operator on polynomials that maps Lorentzian inputs to Lorentzian outputs and connects to induced polymatroids.
    Newly defined to prove the main theorem; no independent evidence outside the paper is given in the abstract.

pith-pipeline@v0.9.0 · 5392 in / 1354 out tokens · 33874 ms · 2026-05-08T16:09:50.362231+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

  1. [1]

    The L ee- Y ang and P \'olya- S chur programs

    Julius Borcea and Petter Br\"and\'en. The L ee- Y ang and P \'olya- S chur programs. I . L inear operators preserving stability. Invent. Math. , 177(3):541--569, 2009

    work page 2009

  2. [2]

    Lorentzian polynomials

    Petter Br \"a nd \'e n and June Huh. Lorentzian polynomials. Annals of Mathematics , 192(3):821--891, 2020

    work page 2020

  3. [3]

    Oxley, Alan D

    Young-Bin Choe, James G. Oxley, Alan D. Sokal, and David G. Wagner. Homogeneous multivariate polynomials with the half-plane property. Advances in Applied Mathematics , 32(1--2):88--187, 2004

    work page 2004

  4. [4]

    Youngbin Choe and David G. Wagner. Rayleigh M atroids. Combinatorics, Probability and Computing , 15(5):765–781, 2006

    work page 2006

  5. [5]

    Intersection T heory of P olymatroids

    Christopher Eur and Matt Larson. Intersection T heory of P olymatroids. International Mathematics Research Notices , 2024(5):4207--4241, March 2024

    work page 2024

  6. [6]

    Linear operators preserving volume polynomials , url =

    Lukas Grund, June Huh, Mateusz Micha ek, Hendrik S \"u ss, and Botong Wang. Linear operators preserving volume polynomials. arXiv preprint , arXiv:2506.22415, 2025. Preprint, 27 pp

    work page arXiv 2025

  7. [7]

    Volume polynomials

    June Huh. Volume polynomials. 2026

    work page 2026

  8. [8]

    Operations on M -convex functions on jump systems

    Yusuke Kobayashi, Kazuo Murota, and Ken'ichiro Tanaka. Operations on M -convex functions on jump systems. SIAM J. Discrete Math. , 21(1):107--129, 2007

    work page 2007

  9. [9]

    Colin J. H. McDiarmid. Rado's theorem for polymatroids. Math. Proc. Cambridge Philos. Soc. , 78(2):263--281, 1975

    work page 1975

  10. [10]

    Dominic J. A. Welsh. Matroid theory . L. M. S. Monographs, No. 8. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976

    work page 1976