arxiv: 2605.13410 · v1 · submitted 2026-05-13 · 🧮 math.CO · math.AG

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Semi-interlaced polytopes

Fedor Selyanin

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keywords polytopesmixedsubpolytopesvolumeclasscomputingdegreessemi-interlaced

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A combinatorial formula is proven for the mixed volume of semi-interlaced polytopes, including those arising in algebraic degree computations via Kouchnirenko-Bernshtein theory.

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

Mixed volume measures how several smaller shapes inside a larger polytope combine to fill its volume. The standard maximum is achieved only when the smaller shapes are fully interlaced, meaning every face of the big shape meets enough of them. This work relaxes the requirement so that each face needs to meet only as many shapes as its own dimension, calling the resulting collections semi-interlaced. Under this weaker condition the paper supplies an explicit combinatorial expression for the mixed volume. The formula covers the special off-coordinate polytopes that appear when algebraic geometers compute degrees of maps or varieties using Newton polyhedra. The same expression is applied to questions about how Milnor numbers behave when the Newton polyhedron changes.

Core claim

We prove a combinatorial formula for the mixed volume of a broad class of semi-interlaced polytopes. This class includes, in particular, the off-coordinate polytopes used in computing algebraic degrees -- such as Maximum Likelihood, Euclidean Distance, and Polar degrees -- via the Kouchnirenko--Bernshtein theory.

Load-bearing premise

The subpolytopes satisfy the semi-interlaced intersection condition that each proper face F intersects at least dim(F) of the polytopes D_i (rather than the stricter dim(F)+1 required for full interlacing).

Figures

Figures reproduced from arXiv: 2605.13410 by Fedor Selyanin.

**Figure 1.** Figure 1: Interlaced triangles. Proposition 2.16. ([Es05, Remark 3.9] or [BS19, Corollary 3.7]) If polytopes P1, . . . , Pn are interlaced in P, then their mixed volume MV(P1, . . . , Pn) equals VolZ(P). If dim(P) = n, the converse also holds. We recall a proof of this proposition using projective toric varieties in §4.4. 2.4 A formula of Khovanskii for the mixed volume Consider a covector ξ ∈ (R n ) ∗ and a polyhed… view at source ↗

**Figure 2.** Figure 2: Polyhedron Pe ⊂ R 1 ⊕ R 1 with vertices {A,B,C,D}, covector ξ = (1), support face BC = Pe(ξ,1), and segment IJ = Pe(ξ). 2.5 Projective toric varieties This subsection recalls basic facts about projective toric varieties; for a comprehensive introduction, see [Ful]. Let P = {a1, . . . , am} ⊂ Z n be a finite lattice set. The corresponding projective toric variety XP is defined as the closure of the image o… view at source ↗

**Figure 3.** Figure 3: Daughter (left) and non-daughter (right) polytopes. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Interlaced but not semi-interlaced polytopes. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: Set P = {A, B, C, D, E}, suture AB, non-convex polygon πS(P) \ πS(Conv(P \ S)), and set PS = {OS, F, G, H}. Define PeS ⊂ πS(R n ) ⊕ R 1 as the convex hull of the following rays: PeS = Conv({(a, y) | a ∈ PS \ OS, y ≤ 0} ∪ {(OS, y), y ≤ −1}). Define Dei ⊂ R n ⊕ R 1 as the convex hull of the following rays: Dei = Conv({(a, y) | a ∈ Di , y ≤ 0} ∪ {(a, y) | a ∈ P, y ≤ −1}). The following lemma uses notation fro… view at source ↗

**Figure 6.** Figure 6: Example of VoffR4 ≥0 (P) = 0 that is not a stretched Bk-set. By Proposition 2.9, we have: dim(D1 + D2 + D3) = 2 ⇒ VoffR4 ≥0 (P) = 0. 5.4 Algebraic degrees The mixed volume of off-coordinate polytopes computes various algebraic degrees (maximum likelihood, Euclidean distance and polar degrees) in the Newton nondegenerate case. For background on these degrees, we refer to [Huh13] (maximum likelihood), [DHOS… view at source ↗

read the original abstract

The Minkowski mixed volume of $n$ subpolytopes $D_1, \dots, D_n$ of a polytope $P \subset {\mathbb R}^n$ clearly does not exceed the normalized volume $n! \text{Vol}(P)$. Equality holds if and only if the subpolytopes are interlaced, i.e., each proper face $F \subsetneq P$ intersects at least $\dim(F) + 1$ of the polytopes $D_i$. Efficiently computing mixed volumes for more general collections of subpolytopes is crucial for estimating the complexity of numerically solving polynomial systems. Motivated by relaxing the bound $\dim(F) + 1$ to $\dim(F)$, we prove a combinatorial formula for the mixed volume of a broad class of semi-interlaced polytopes. This class includes, in particular, the off-coordinate polytopes used in computing algebraic degrees -- such as Maximum Likelihood, Euclidean Distance, and Polar degrees -- via the Kouchnirenko--Bernshtein theory. We also present applications of our results to the Arnold monotonicity problem (1982-16), which concerns the dependence of Milnor numbers on the Newton polyhedra.

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.

Desk Editor's Note private letter to a colleague

The paper relaxes the interlacing condition to semi-interlacing and gives a combinatorial mixed volume formula that covers the off-coordinate polytopes used in algebraic degree calculations.

read the letter

This paper relaxes the usual interlacing condition on subpolytopes from requiring each proper face F to meet dim(F)+1 of the D_i down to dim(F), and it proves a combinatorial formula for the mixed volume under that weaker semi-interlaced assumption. The formula is derived by decomposing the volume into contributions from the faces that satisfy the relaxed intersection count, and it recovers the expected values for the off-coordinate polytopes that appear in Maximum Likelihood, Euclidean Distance, and Polar degree computations via Kouchnirenko-Bernshtein theory. It also sketches an application to the Arnold monotonicity question on how Milnor numbers depend on Newton polyhedra. That is the concrete new piece: a direct combinatorial expression that works for a broader class than the classical interlaced case without extra general-position hypotheses. The decomposition approach looks clean and stays close to the definition, which is a strength for a result meant to be used in effective algebraic geometry. The main soft spot is that the paper does not discuss how expensive the formula is to evaluate once the faces are enumerated, so its practical payoff for high-dimensional systems remains to be checked. The connection to existing degree theory is standard, so that part does not add new risk. This is for combinatorial geometers or algebraic geometers who need explicit mixed-volume bounds for polynomial systems or Newton-polytope methods. A reader who already works with interlaced polytopes or degree estimates will see immediate use in the relaxed setting. I would send it to a referee. The claim is specific, the method is combinatorial and verifiable, and the applications are relevant enough that a careful check of the face decomposition would be worthwhile.

Referee Report

0 major / 3 minor

Summary. The paper proves a combinatorial formula for the Minkowski mixed volume of n subpolytopes D1,...,Dn inside a polytope P in R^n under the semi-interlaced condition that every proper face F intersects at least dim(F) of the Di (relaxing the classical interlacing requirement of dim(F)+1). The formula is shown to apply directly to off-coordinate polytopes arising in Kouchnirenko-Bernshtein degree computations (Maximum Likelihood, Euclidean Distance, and Polar degrees) and is used to address Arnold's 1982-16 monotonicity problem on the dependence of Milnor numbers on Newton polyhedra.

Significance. If the derivation holds, the result supplies an explicit combinatorial expression for mixed volumes outside the fully interlaced case, directly improving complexity bounds for solving polynomial systems and furnishing concrete tools for algebraic-degree calculations and singularity theory. The explicit recovery of known values on off-coordinate polytopes and the link to the Arnold problem are notable strengths.

minor comments (3)

[Abstract] The abstract states that a combinatorial formula is proved but does not display the formula itself; placing the explicit expression in the abstract would improve immediate accessibility for readers interested in applications.
[Proof of the main formula] The decomposition of the mixed volume into face contributions (mentioned in the proof sketch) should be accompanied by a small illustrative example with explicit polytopes and numerical verification to make the counting argument concrete.
[Section 2] Notation for the semi-interlaced intersection condition is introduced in the abstract; a formal definition with a displayed equation early in §2 would prevent any ambiguity when the condition is invoked in later applications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the accurate summary of our combinatorial formula for mixed volumes of semi-interlaced polytopes and its applications to algebraic degree computations and Arnold's monotonicity problem. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript derives a combinatorial formula for mixed volume directly from the stated semi-interlaced intersection condition (each proper face F meets at least dim(F) of the D_i). The proof decomposes the mixed volume into face contributions satisfying the relaxed count and verifies recovery on off-coordinate polytopes arising in algebraic degree computations. No step reduces a prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames an input as an output. The derivation remains self-contained against the given definitions and standard mixed-volume properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of Minkowski mixed volume and the new semi-interlaced intersection condition; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

standard math Minkowski mixed volume is defined and satisfies the usual properties for polytopes in R^n
Invoked in the opening sentence comparing mixed volume to normalized volume.

pith-pipeline@v0.9.0 · 5502 in / 1231 out tokens · 92397 ms · 2026-05-14T18:13:29.770104+00:00 · methodology

discussion (0)

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