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Polynomials over idempotent semifields

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Pith's one-line read Every complete idempotent semifield is algebraically closed: every polynomial function of positive degree factors into linear terms.

desk verdict Solid, fully proved extension of tropical polynomial factorization to non-totally-ordered commutative idempotent semifields; complete ones are algebraically closed. read the letter →

arxiv 2607.10492 v1 pith:5IEGUFSF submitted 2026-07-11 math.AC

Paul Poncet This is my paper · ORCID
classification math.AC MSC 06F0512K1016Y60
keywords idempotentsemifieldstropicalalgebrapolynomialfactorizationalgebraicallyclosedclosablepolynomialsradicabilitypreradicabilitymaxpolynomials
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper develops factorization for univariate polynomials over idempotent semifields without assuming total order. It separates formal polynomials from the functions they induce and proves two fundamental theorems: a polynomial is closed precisely when it splits into ordered linear factors (its corners), and it is closable precisely when the induced function splits. Algebraic closedness is therefore equivalent to every polynomial being closable, which immediately implies that every complete idempotent semifield is algebraically closed. The same circle of ideas links algebraic closedness to the weaker notions of preradicability and radicability and shows how those properties control the existence of solutions to polynomial inequalities and equations. The work therefore supplies a usable algebraic closedness theory for the general, non-totally-ordered case that appears under products and quotients.

What carries the argument

The closure of a closable polynomial: the greatest polynomial that induces the same function, obtained by taking the infima of the scaled values of the function. Closed polynomials are exactly those equal to their own closure; they coincide with the polynomials that split into linear factors ordered by their corners.

What would settle it

Exhibit a complete non-commutative or finite idempotent semifield in which some positive-degree polynomial function fails to split into linear factors, or construct a non-closable polynomial over a complete commutative infinite idempotent semifield.

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Extended reading notes

Core claim

An idempotent semifield is algebraically closed (every polynomial function of positive degree splits into linear factors) if and only if every polynomial is closable; in particular every complete idempotent semifield is algebraically closed. Equivalently, a polynomial is closed if and only if it factors as a product of linear terms with ordered corners, and closable if and only if its induced function admits such a factorization.

Load-bearing premise

Every semifield in the paper is assumed commutative and infinite; both hypotheses are used for the modular identity, Frobenius identity, and the existence of the needed infima, and the statements can fail for the Boolean semifield.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper develops a factorization theory for univariate polynomials over commutative infinite idempotent semifields without assuming total order. It introduces closable and closed polynomials via infima of the form ∧_x bp(x)x^{-j}, proves that a polynomial is closed iff it splits into ordered linear factors (First Fundamental Theorem 5.4), and closable iff its associated polynomial function splits (Second Fundamental Theorem 5.12). Algebraic closedness (every positive-degree polynomial function splits) is thereby characterized as every polynomial being closable; in particular every complete idempotent semifield is algebraically closed. The paper further relates algebraic closedness to preradicability and radicability, obtains existence results for polynomial inequalities and equations, and shows that radicability yields an isomorphism between rational polynomials and polynomial functions together with splitting of every rational polynomial.

Significance. The work removes the total-order hypothesis that has dominated the tropical/max-plus literature (Cuninghame-Green–Meijer, Baccelli et al., Butkovič, Castella, Rump) while recovering and extending the classical factorization theorems. The clean characterization that completeness implies algebraic closedness, the ordered-corner uniqueness lemma, and the precise hierarchy equationally closed ⇒ radicable ⇒ algebraically closed + order-dense ⇒ algebraically closed ⇒ preradicable (with total-order collapses) are substantial contributions. The treatment of non-unique baskets of roots, completion of partial baskets, and the radicable-closure construction are technically solid and fill a genuine gap. The results are self-contained under the stated standing hypotheses and should become a standard reference for polynomials over general idempotent semifields.

minor comments (6)
  1. §3.1 disclaimer: the global restriction to commutative infinite semifields is essential and correctly flagged, but a short parenthetical reminder at the first use of the modular identity (2) and of Frobenius (Lemma 2.5) would help readers who skip the disclaimer.
  2. Lemma 5.2 (uniqueness of ordered corners): the induction step when bp(0)=0 is dispatched by dividing by X^{val p}; a one-line justification that the resulting corners remain ordered would make the argument fully explicit.
  3. Example 5.3 / Figure 1: the directed graph of baskets is helpful; if the journal permits, a small TikZ rendering would improve readability over the pure text description.
  4. Notation: the hat notation bp for the polynomial function is consistent with Baccelli et al., but the closure p (overline) is occasionally hard to distinguish in plain text; consider a bold or calligraphic alternative in the final version.
  5. §7.4 rational polynomials: the isomorphism of Proposition 7.18 is stated cleanly; a brief remark that the construction recovers Castella’s totally-ordered case would orient the reader.
  6. References: the arXiv preprints of Akian–Bapat–Gaubert and Tolliver are cited; if published versions now exist they should be updated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: closed/closable defined via independent infima construction; splitting equivalences derived by explicit corner computation and induction, not by redefinition.

full rationale

The paper's central results (First/Second Fundamental Theorems 5.4/5.12 equating closed/closable polynomials with splitting of p/bp into ordered linear factors, and Corollaries 5.14–5.15 that algebraic closedness is equivalent to universal closability and that completeness implies algebraic closedness) rest on self-contained definitions and proofs. Closable is defined (Sec. 4.1, Eq. 8) purely as existence of the infima V{bp(x)x^{-j}:x∈k∗}, with the closure p-bar formed from those coefficients (reminiscent of a multiplicative Legendre transform but not presupposing factorization). Closed means p=p-bar. The proofs then show, via Crosby residuation (Thm 3.3), modular/Frobenius identities (Eqs. 2, Lem. 2.5), ordered-basket uniqueness by induction (Lem. 5.2), and concavity (Thm 5.4), that these properties are equivalent to the existence of corners c_k=p_{n-k}p_{n-k+1}^{-1} yielding the product factorization. Completeness supplies the infima by definition of the lattice, so Cor. 5.15 follows immediately without assuming the conclusion. No parameter fitting, no self-citation that is load-bearing for the equivalences (citations to Rump/Castella/Baccelli/Cuninghame-Green are extensions of prior special cases, not uniqueness theorems imported to force the present claims), and no renaming of known results. The standing hypotheses (commutative+infinite, §3.1) are explicit and used openly for the modular identity and exclusion of B; they do not create a definitional loop. The derivation chain is therefore independent of its conclusions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The paper is pure algebra. Load-bearing background is standard semiring/lattice-group theory plus the standing restriction to commutative infinite idempotent semifields. No fitted parameters. Invented entities are definitional (closable/closed, baskets of corners) with independent mathematical content via the fundamental theorems.

assumptions (4)
  • domain assumption All idempotent semifields considered are commutative and infinite (≠ Boolean semifield B).
    Global disclaimer §3.1; used for modular identity, Frobenius, residuation statements, and Lemma 2.8 on inf k*.
  • standard math Idempotent semifields induce lattice-ordered groups on k* with modular identity xy=(x∨y)(x∧y) and distributivity of multiplication over arbitrary sups/infs.
    Recalled in §2.3 from Birkhoff/Bigard–Keimel–Wolfenstein; used throughout factorization and closure arguments.
  • standard math Crosby's residuation theorem: for q≠0 there exists p/q with f·q ≤ p ⇔ f ≤ p/q, given by the explicit inf formula (7).
    Cited as Crosby Thm 5.2.3; used to make k[X] residuated and to prove quotients of closed polynomials remain closed (Lem 5.9).
  • standard math Frobenius identity (x+y)^n = x^n + y^n in ×-cancellative commutative idempotent semirings.
    Lemma 2.5; used for convexity of polynomial functions and for k-congruence of rational polynomials.
invented entities (3)
  • Closable / closed polynomial (infima of bp(x)x^{-j}) independent evidence
    purpose: Canonical maximal representative of a polynomial function; intermediate notion between formal polynomials and functions that enables factorization theorems without total order.
    Defined in §4; independent content via First/Second fundamental theorems equating closedness/closability with splitting.
  • Basket of corners of a closed polynomial independent evidence
    purpose: Unique nonincreasing full basket of roots used to write the linear factorization.
    Lemma 5.2 uniqueness; used as the explicit factors in both fundamental theorems.
  • Preradicability (maps x ↦ x^k residuated) independent evidence
    purpose: Weaker-than-radicable condition guaranteeing solutions to certain polynomial inequalities.
    Introduced §6; linked to algebraic closedness by Prop 6.4 and Thm 6.6.

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Pith. "Pith review of Polynomials over idempotent semifields." pith.science (2026). https://pith.science/paper/5IEGUFSF

@misc{pith2026260710492,
  author       = {Pith},
  title        = {Pith review of: Polynomials over idempotent semifields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5IEGUFSF}},
  note         = {Machine review of arXiv:2607.10492}
}
read the original abstract

We study univariate polynomials with coefficients in an idempotent semifield and their factorization. We do not assume the idempotent semifield under consideration to be totally ordered, in contrast with most of the existing work on this topic. We notably determine when a polynomial splits into linear factors, and when its associated polynomial function does so. These results lead us to characterize algebraically closed idempotent semifields -- those in which every polynomial function splits. We prove in particular that every complete idempotent semifield is algebraically closed. We also relate algebraic closedness to the properties of preradicability and radicability and to the existence of solutions to polynomial equations or inequalities.

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