How to stab a polytope

Francesca Zaffalon; Sebastian Seemann

REVIEW 5 major objections 4 minor 21 references

How to stab a polytope

T0 review · 5 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Chow-form signs decide which subspaces stab a polytope

desk verdict A promising framework for stabbing sets via Chow-form sign conditions, but the proofs as written leave load-bearing gaps in the deformation arguments. read the letter →

arxiv 2412.00551 v1 pith:Y52FI7S2 submitted 2024-11-30 math.CO math.AG

Sebastian Seemann , Francesca Zaffalon This is my paper · ORCID

classification math.COmath.AG MSC 52B1114M1514P10

keywords polytopeGrassmannianChowformSchubertarrangementstabbingsetsemialgebraicamplituhedrontotalpositivity

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 establishes that the set of all $k$-dimensional linear subspaces of projective space that intersect a given full-dimensional polytope $P$ can be described completely by the signs of the Chow forms of the faces of $P$ of complementary dimension. More precisely, the authors define the $k$-face Schubert arrangement of $P$ and prove that each stabbing chamber, consisting of subspaces meeting the same family of $(n-k)$-dimensional faces, is exactly a fixed sign pattern of the Chow forms bounding those faces. They then show that the full stabbing set $P^{[k]}$ is the closure of its maximally stabbing subset and is cut out by finitely many sign inequalities, one per boundary of some $(n-k)$-face, giving an explicit semialgebraic description of $P^{[k]}$ inside the Grassmannian. The result reduces a geometric incidence question to a finite sign computation, and it recovers known slicing descriptions while connecting to amplituhedra.

What carries the argument

The central object is the $k$-face Schubert arrangement $H^k_P$ of a polytope $P$, the finite collection of Schubert divisors in $\operatorname{Gr}(k,n)$ associated to the linear spans of the $(n-k-1)$-dimensional faces of $P$. Each divisor is the zero set of the Chow form of one such face, so the arrangement is packaged as a vector $C^k_P = (C_{G_1},\dots,C_{G_f})$ of Chow forms; evaluated on a subspace $V$, this vector records, up to a common scalar, which face spans $V$ intersects. The load-bearing mechanism is the correspondence between sign vectors of this Chow-form vector and stabbing behaviour: within a stabbing chamber the sign pattern on the boundaries of the determining faces is constant, and a subspace intersects $P$ exactly when its sign pattern can be obtained from a maximally stabbing pattern by setting some entries to zero. The argument runs through straight-line deformations of one intersection point to another inside a face, along which the Chow forms vary linearly in the Plücker coordinates, so signs can change only when the moving point crosses a facet.

What would settle it

Take a small polytope, such as the octahedron in $\mathbb{P}^3$ treated in Example 4.4, and enumerate the sign vectors of the Chow-form vector over $\operatorname{Gr}(k,n)$; if one finds a subspace $V$ whose sign vector on the boundaries of some face $F$ satisfies $\operatorname{sign}(C^k_P(V)|_F) \le \operatorname{sign}(C^k_P(W)|_F)$ for a maximally stabbing $W$ but $V$ does not meet $P$, then Theorem 4.1 is false. A more targeted check is to follow the deformation paths constructed in the proofs of Theorem 3.1 and Proposition 3.7 and record whether the matrix rank ever drops or whether a Chow form changes sign at a moment when the moving point is not crossing a facet of the relevant face.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is Theorem 4.1: a $k$-dimensional subspace $V$ of an $n$-dimensional real vector space intersects the polytope $P$ if and only if there exists a maximally stabbing subspace $W \in P^{[k]}_{\max}$ and an $(n-k)$-dimensional face $F$ of $P$ such that the sign vector of the Chow forms of $P$ evaluated at $V$, restricted to the boundary facets of $F$, is componentwise less than or equal, up to overall sign, to the sign vector evaluated at $W$: $\operatorname{sign}(C^k_P(V)|_F) \le \operatorname{sign}(C^k_P(W)|_F)$. The paper proves this by showing that the stabbing chambers of $P^{[k]}_{\max}$ are exactly the chambers of the $k$-face Schubert arrangement, that $P^{[k]}$ is the closure of $P^{[k]}_{\max}$, and that passing from maximally stabbing subspaces to all stabbing subspaces is captured by adding zeros to the sign vector. For $(n-k)$-simplicial polytopes, including cyclic polytopes, the description specializes to an explicit alternating sign condition on the Chow forms of a face, and the paper identifies the totally non-negative Grassmannian as lying in the closure of a single stabbing chamber of the standard simplex.

Load-bearing premise

The load-bearing premise is that when one intersection point of the subspace with a face is moved in a straight line to another point inside that same face, the moving family stays a genuine $k$-dimensional subspace and the signs of the relevant Chow forms change only when the point crosses a facet of that face; if a rank drop or an off-facet sign change occurs somewhere along the path, the chamber characterization would need extra hypotheses.

Editorial extensions

If this is right

The $k$-stabbing set $P^{[k]}$ is a semialgebraic subset of $\operatorname{Gr}(k,n)$ defined by explicit sign inequalities, so membership of a subspace can be decided by evaluating finitely many Chow forms.
For hyperplanes ($k=n-1$) the construction recovers the slicing chambers of a polytope, and for points ($k=1$) it recovers the ordinary facet description of a polytope.
For $(n-k)$-simplicial polytopes, including cyclic polytopes, intersection with a given face is characterized by an alternating sign pattern $\operatorname{sign}(C_{S\setminus s_{n-k+1}}(V),\dots,C_{S\setminus s_1}(V)) \equiv (+,-,\dots,(-1)^{n-k})$, giving a direct computational test.
The totally non-negative Grassmannian lies in the closure of the stabbing chamber of the standard simplex determined by consecutive faces, and the amplituhedron sits inside the stabbing set of a cyclic polytope, connecting the sign description to amplituhedron sign-flip characterizations.
The paper's Conjecture 3.5 predicts that for polytopal Schubert arrangements the number of regions inside $P^{[k]}$ equals the number of realized sign vectors, which would make the chamber count a purely combinatorial invariant of the polytope.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Theorem 4.1 holds, the same sign-vector description should extend to any convex body whose boundary is a finite union of algebraic hypersurface pieces, since the proof uses only convexity and linearity of the deformation; testing this on ellipsoids or spectrahedra would be a natural stress test.
The realizability problem for sign vectors of $C^k_P$ is an oriented-matroid-type invariant of a polytope; for simplices the paper already notes the connection, so one could investigate whether the set of realized sign vectors determines the combinatorial type of $P$.
The rank assumptions in the deformation argument are the part most likely to hide a counterexample; a computational search on low-dimensional polytopes for straight-line paths that drop rank or change a Chow-form sign off a facet would either confirm the gap or localize it.
Procedure 4.2 gives an explicit inequality description that could serve as a membership oracle for the loop amplituhedron projection, where the paper only records the inclusion; checking whether the sign inequalities characterize the image is a concrete next step.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague, here's my read on Seemann-Zaffalon.

The paper's core idea is good and genuinely new. They associate to a polytope a Schubert arrangement in Gr(k,n) defined by Chow forms of codimension-(n-k-1) faces, and claim that the k-stabbing set P[k] is cut out by sign inequalities on Chow forms evaluated on boundaries of (n-k)-dimensional faces. This unifies the hyperplane slicing chambers of Brandenburg-De Loera-Meroni with the k>1 case and makes a clean connection to amplituhedra. The statement of Theorem 4.1 is appealing and would be a real contribution if it held.

What I like: the setup (Setup 2.7) is careful about sign vectors modulo overall scalar; the examples, especially Example 4.4 with the octahedron, are worked out concretely; Section 5 gives a clean sign characterization for simplicial polytopes that reduces to the usual hyperplane separation condition for k=n-1. The citations to the amplituhedron and Schubert arrangement literature look right, and the paper is honest about where it connects to earlier work.

The soft spots are real, though. The stress-test note is on target: in Theorem 3.1, the path U^1_t replaces one row by (1-t)v_1 + t w_1 and claims 'by convexity' the matrix stays full-rank for all t. Convexity only keeps the new row inside the face; it does nothing to prevent the k rows from becoming dependent. Without that, the Plücker coordinates aren't defined along the path, and the sign-preservation argument collapses. The same gap propagates into Proposition 3.7 and Theorem 4.1. Lemma 2.8's non-vanishing proof is also not rigorous as written; the induction step is asserted more than shown. And the converse direction of Theorem 3.1 skips the case where the moving row crosses an extra face not in FP(V). Proposition 3.7 and Theorem 4.1 do refer to each other in a way that is circular; the 'by proof of Theorem 4.1' in 3.7 and the 'arguments similar to Theorem 3.7' in 4.1 need disentangling. Procedure 4.2 assumes the chosen v_F span a maximally stabbing subspace; not proven.

None of this makes me think the main claim is false; the framework is plausible and probably salvageable. But as written the derivation does not support the theorems. There is no code or formalization, so the gaps are not compensated empirically.

Who benefits: researchers in geometric combinatorics, positive geometry, and the amplituhedron literature. It deserves a serious referee, but the referee should be asked to check the deformation arguments carefully. My recommendation: send to review, and expect major revision. For my own work, I wouldn't cite it until the path argument is fixed.

Best.

Referee Report

5 major / 4 minor

Summary. The paper studies the set P[k] of k-dimensional linear subspaces of R^n that intersect a full-dimensional polytope P in P^{n-1}. The authors introduce the k-face Schubert arrangement of P, defined by Chow forms of (n-k-1)-faces, and claim that stabbing chambers of P[k]_max are characterized by the signs of the Chow forms of the boundaries of the intersected (n-k)-faces (Theorem 3.1). They further claim that P[k] is the closure of P[k]_max (Proposition 3.6), that V ∈ P[k] if and only if sign(C^k_P(V)) ≤ sign(C^k_P(W)) for some W ∈ P[k]_max (Proposition 3.7), and that membership in P[k] is equivalent to a sign inequality on the boundary Chow forms of a single face (Theorem 4.1). A procedure for producing inequalities, a result for simplicial polytopes, and connections to amplituhedra are also presented.

Significance. The problem of describing which k-subspaces stab a polytope is natural and connects to hyperplane arrangements in Grassmannians, slicing chambers, and the amplituhedron program. If the sign-characterization results were rigorously established, they would provide a useful semialgebraic description of P[k] via Chow forms, and the paper includes instructive examples (octahedron, simplex) and a concrete conjecture. The paper is clearly written and the examples are helpful. However, the central proofs contain several unresolved technical gaps, so the main theorems are not yet proven in the submitted form.

major comments (5)

[§3.1, proof of Theorem 3.1] The deformation argument is not justified. For U^1_t built by replacing the first row of a k×n matrix with (1-t)v_1 + t w_1, convexity of P only implies that the new row stays in the face F_1; it does not imply that the k rows remain linearly independent for every t ∈ [0,1]. If a maximal minor vanishes at some t0, then U^1_{t0} is not a point of Gr(k,n), and the Plücker coordinates, Chow form evaluations, and their signs are undefined along the path. The text asserts that the map t ↦ U^1_t is a well-defined path in Gr(k,n) and that Plücker coordinates are linear in t, but no rank-preservation proof is given. This gap afflicts the forward and converse directions of Theorem 3.1 and is reused in Proposition 3.7 and Theorem 4.1; a rank drop would break the claimed chamber characterization.
[§2.2, Lemma 2.8] The induction proof of non-vanishing of C^k_P(V) is incomplete. The sentence 'Since V is not contained in span(F) for every (n-k-1)-dimensional face F, there exists F′ such that V ∩ span(F) has dimension smaller than k' is not a valid inference as written; moreover, V ∩ span(F′) has dimension strictly less than k, so the expression C^{F′}_k(V ∩ span(F′)) is not defined, since the Chow forms in Lemma 2.8 are evaluated on k-dimensional subspaces. The induction hypothesis cannot be applied to a lower-dimensional intersection. The base case is also stated too tersely. Since Lemma 2.8 is later used in the proof of Proposition 3.7, a correct proof of the non-vanishing assertion is needed.
[§3.2 and §4, Proposition 3.7 and Theorem 4.1] There is a circular dependency between Proposition 3.7 and Theorem 4.1. The proof of Proposition 3.7 sends the case of equal nonzero boundary sign vectors to 'by proof of Theorem 4.1', while the first line of the proof of Theorem 4.1 invokes 'Theorem 3.7' (i.e., Proposition 3.7) for the forward direction. Moreover, the forward direction of Proposition 3.7 is asserted without proof; if it is intended to follow from Proposition 3.6, that argument is not supplied, and the proof of Proposition 3.6 itself needs to justify the existence of k linearly independent points in V ∩ P in the first case. The logical order of the results must be reworked so that the two statements are not mutually referential.
[§4, proof of Theorem 4.1, converse direction] The converse direction of Theorem 4.1 is not established. The construction of the matrices U^i_t inherits the rank-drop problem of Theorem 3.1. In addition, the assertions 'Along this path there exists i0 ∈ {0,...,k} and t0 such that U^i_t intersects P for t < t0 and i ≤ i0 and U^i_t does not intersect P for t > t0 and i ≥ i0' and 'the spaces along the path U^i_t can only exit the polytope through the face F' are not derived. The intersection of a moving linear subspace with a polytope need not change monotonically in t, and a Chow form of a boundary of F could vanish or change sign without the subspace exiting P through F, or vice versa. This part requires a rigorous path argument.
[§4, Procedure 4.2] The procedure assumes without proof that the witness subspace V_{F_1,...,F_k} = span(v_{F_1},...,v_{F_k}) is a point of Gr(k,n) and belongs to P[k]_max. The condition that F_1,...,F_k are 'not contained in a k-dimensional face' does not guarantee that the vectors v_{F_i} (sums of vertices) are linearly independent, nor that W stabs P maximally, nor that the sign vector of C^k_P(W) restricted to F_i is the correct reference for all V intersecting F_i. Consequently the inequality description in step 2 is not justified by Theorem 4.1 as stated.

minor comments (4)

[§3.2, proof of Proposition 3.7] The step 'Since by Lemma 2.8, the vector of Chow form is not zero, we can choose a face F of P such that C^k_P(V) is not zero on every boundary of F' does not follow: a non-vanishing vector of Chow forms may have a zero entry on a boundary of every (n-k)-face. A separate argument is needed for the existence of such an F.
[§4, Theorem 4.1] There are cross-reference errors: Theorem 4.1 cites 'Theorem 3.7' for what is Proposition 3.7, and the reference to the construction in Theorem 3.1 should be made explicit when it is imported into other proofs.
[§2.2, Setup 2.7] The choice of n-k 'linearly independent vertices' of an (n-k-1)-face requires justification: affinely independent vertices of a face need not be linearly independent as vectors in R^n after choosing representatives. The authors should specify a convention, for example by working in an affine chart or with the cone over P.
[§4, Example 4.4] The colors mentioned in the sign vector are irrelevant for a printed version, and the statement that the description extends to every facet should be made precise by listing the reference signs for the other facets.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sign-characterization results are substantive geometric claims, not consequences of their own definitions; the proof gaps are correctness issues, not circular reductions.

full rationale

The paper's claimed derivation is self-contained in the sense required by a circularity audit: none of the main results is equivalent to its inputs by construction. Setup 2.7 defines Chow forms of faces as linear forms in Plücker coordinates, while Theorem 3.1, Proposition 3.7, and Theorem 4.1 assert nontrivial equivalences between geometric intersection with a polytope and sign conditions on those Chow forms; the proofs do not simply restate the definitions. There is no fitted parameter renamed as a prediction, no load-bearing self-citation (the authors' own prior work is not cited for the central claims), no imported uniqueness theorem from the authors, and no ansatz smuggled in via citation. The cross-references between Proposition 3.7 and Theorem 4.1 (Prop 3.7 says one case follows 'by proof of Theorem 4.1'; Theorem 4.1 says the forward direction follows 'by Theorem 3.7') are not a circular dependency in the implication graph: Proposition 3.7's use of Theorem 4.1 is an instance of Theorem 4.1's independently proved converse, and Theorem 4.1's forward direction rests on Proposition 3.7's forward assertion, which can be derived from Proposition 3.6. The main weakness is a correctness gap, not circularity: the deformation paths U^i_t in Theorem 3.1 are asserted to lie in Gr(k,n) 'by convexity of the polytope' without proving that the rows remain linearly independent, so the sign-preservation argument is unsupported; that is a mathematical gap, not a reduction of the conclusion to the hypothesis.

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

The paper's main results depend on standard Chow form and Grassmannian machinery plus a few ad hoc technical assumptions: the non-vanishing of the Chow vector, the full-rank preservation of interpolating matrices, and the existence of suitably generic witness subspaces in Procedure 4.2. There are no fitted parameters or independently postulated entities.

assumptions (4)

domain assumption All polytopes are full-dimensional in P^{n-1} over the real field, and dimensions are counted in affine space.
Stated in Section 2.1 and Definition 2.4; the entire sign-based framework requires real signs and full-dimensional polytopes.
ad hoc to paper The vector of Chow forms C^k_P(V) is never the zero vector, so sign vectors are well-defined projectively (Lemma 2.8).
The provided induction proof has a gap ('Since V is not contained in span(F)... there exists a face F'...') and does not rigorously establish the statement; this is load-bearing for defining sign vectors.
ad hoc to paper The interpolating matrices U_t built by replacing rows with convex combinations (1-t)v1 + t w1 have full rank k for every t, and the relevant Chow forms change sign only at explicit boundary crossings.
Invoked in the proofs of Theorem 3.1 and Proposition 3.7 ('the map t -> U^1_t is a well-defined path in Gr(k,n)', 'By convexity of the polytope') but never proven; a rank drop would invalidate the path arguments.
domain assumption Known results on total positivity and amplituhedra, including Karp's sign variation theorem [13, Theorem 1.6] and Lam's well-definedness of Z_tilde [15], are accepted.
Used in Section 2.3 and Proposition 5.4 to connect stabbing sets with totally nonnegative Grassmannians and cyclic polytopes.

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Pith. "Pith review of How to stab a polytope." pith.science (2026). https://pith.science/paper/Y52FI7S2

@misc{pith2026241200551,
  author       = {Pith},
  title        = {Pith review of: How to stab a polytope},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y52FI7S2}},
  note         = {Machine review of arXiv:2412.00551}
}

read the original abstract

We study the set of linear subspaces of a fixed dimension intersecting a given polytope. To describe this set as a semialgebraic subset of a Grassmannian, we introduce a Schubert arrangement of the polytope, defined by the Chow forms of the polytope's faces of complementary dimension. We show that the set of subspaces intersecting a specified family of faces is defined by fixing the sign of the Chow forms of their boundaries. We give inequalities defining the set of stabbing subspaces in terms of sign conditions on the Chow form.

Figures

Figures reproduced from arXiv: 2412.00551 by the authors.

**Figure 2.** The line V from Example 4.4 stabbing the octahedron and the sign condition on the boundaries of the stabbed faces. Example 4.4. Consider the octahedron P = conv(v12, v13, v14, v24, v23, v34) ⊆ P 3 where vi j = ei + ej , for {ei}i∈[4] standard basis vectors. Fix k = 1, that is, we want to study the set of lines intersecting the octahedron. The first set is to define the Chow forms we are interested in studying. Asso… view at source ↗

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Reference graph

Works this paper leans on

21 extracted references · 16 canonical work pages

[1]

Arkani-Hamed, Y

N. Arkani-Hamed, Y . Bai and T. Lam: Positive Geometries and Canonical Forms Journal of High Energy Physics No. 11 (2017)

work page 2017
[2]

Arkani-Hamed, H

N. Arkani-Hamed, H. Thomas, and J. Trnka: Unwinding the Amplituhedron in Binary, Journal of High Energy Physics No. 16 (2018)

work page 2018
[3]

Arkani-Hamed and J

N. Arkani-Hamed and J. Trnka: The Amplituhedron, Journal of High Energy Physics No. 10 (2014)

work page 2014
[4]

Bezanson, A

J. Bezanson, A. Edelman, S. Karpinski, and V . B. Shah,: Julia: A fresh approach to numerical computing, SIAM review No. 59(1) (2017)

work page 2017
[5]

Brandenburg and J

M. Brandenburg and J. A. De Loera and C. Meroni: The Best Ways to Slice a Polytope, Mathematics of Computations (2024)

work page 2024
[6]

Breiding, B

P. Breiding, B. Sturmfels, and K. Wang: Computing Arrangements of Hypersur- faces, arXiv:2409.09622

arXiv
[7]

Dalbec, and B Sturmfels: Introduction to Chow forms , Invariant Methods in Discrete and Computational Geometry: Proceedings of the Curac ¸ao Conference, 13–17 June, 1994 (1995)

J. Dalbec, and B Sturmfels: Introduction to Chow forms , Invariant Methods in Discrete and Computational Geometry: Proceedings of the Curac ¸ao Conference, 13–17 June, 1994 (1995)

work page 1995
[8]

Dian, and P

G. Dian, and P. Heslop: Amplituhedron-like geometries, Journal of High Energy Physics No. 11 (2021)

work page 2021

Show all 21 references

[9]

Even-Zohar, T

C. Even-Zohar, T. Lakrec, M. Parisi, R. Tessler, M. Sherman-Bennett and L. Williams: Cluster algebras and tilings for the m=4 amplituhedron , arXiv:2310.17727

arXiv
[10]

Fulton: Young tableaux: with applications to representation theory and geom- etry, Cambridge University Press No

W. Fulton: Young tableaux: with applications to representation theory and geom- etry, Cambridge University Press No. 35 (1997)

1997
[11]

Gale: Neighborly and cyclic polytopes, Proc

D. Gale: Neighborly and cyclic polytopes, Proc. Sympos. Pure Math. Vol. 7. No. 7 (1963)

1963
[12]

F. R. Gantmakher and M. G. Krein. Oscillyacionye matricy i yadra i malye kolebaniya mehani ˇceskih sistem Gosudarstv. Isdat. Tehn.-Teor. Lit., Moscow- Leningrad, (1950)

1950
[13]

Karp: Sign variation, the Grassmannian, and total positivity, Journal of Com- binatorial Theory, Series A No

S. Karp: Sign variation, the Grassmannian, and total positivity, Journal of Com- binatorial Theory, Series A No. 145 (2017)

2017
[14]

Karp and L

S. Karp and L. Williams: The m=1 amplituhedron and cyclic hyperplane arrange- ments, International Mathematics Research Notices No. 8 (2019) HOW TO STAB A POLYTOPE 21

2019
[15]

Lam: Totally nonnegative Grassmannian and Grassmann polytopes , Current Developments in Mathematics No.1 (2014)

T. Lam: Totally nonnegative Grassmannian and Grassmann polytopes , Current Developments in Mathematics No.1 (2014)

2014
[16]

Lukowski, M

T. Lukowski, M. Parisi and L. Williams: The Positive Tropical Grassmannian, the Hypersimplex, and the m = 2 Amplituhedron, International Mathematics Research Notices No. 3 (2023)

2023
[17]

Mandelshtam, D

Y . Mandelshtam, D. Pavlov and E. Pratt: Combinatorics of m = 1 Grasstopes, arXiv:2307.09603

arXiv
[18]

Mazzucchelli, D

E. Mazzucchelli, D. Pavlov and K. Wang:Hyperplane Arrangements in the Grass- mannian, arXiv:2409.04288

arXiv
[19]

Parisi, M

M. Parisi, M. Sherman-Bennett and L. Williams: The m = 2 amplituhe- dron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers , arXiv:2104.08254

arXiv
[20]

Ranestad, R

K. Ranestad, R. Sinn, and S. Telen: Adjoints and Canonical Forms of Tree Ampli- tuhedra, arXiv:2402.06527

arXiv
[21]

Sturmfels: Totally positive matrices and cyclic polytopes, Linear Algebra and its Applications No

B. Sturmfels: Totally positive matrices and cyclic polytopes, Linear Algebra and its Applications No. 107 (1988) SEBASTIAN SEEMANN KU Leuven e-mail: sebastian.seemann@kuleuven.be FRANCESCA ZAFFALON MPI for Mathematics in the Sciences, Leipzig Weizmann Institute of Science e-ma...

1988

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Arkani-Hamed, Y

N. Arkani-Hamed, Y . Bai and T. Lam: Positive Geometries and Canonical Forms Journal of High Energy Physics No. 11 (2017)

work page 2017

[2] [2]

Arkani-Hamed, H

N. Arkani-Hamed, H. Thomas, and J. Trnka: Unwinding the Amplituhedron in Binary, Journal of High Energy Physics No. 16 (2018)

work page 2018

[3] [3]

Arkani-Hamed and J

N. Arkani-Hamed and J. Trnka: The Amplituhedron, Journal of High Energy Physics No. 10 (2014)

work page 2014

[4] [4]

Bezanson, A

J. Bezanson, A. Edelman, S. Karpinski, and V . B. Shah,: Julia: A fresh approach to numerical computing, SIAM review No. 59(1) (2017)

work page 2017

[5] [5]

Brandenburg and J

M. Brandenburg and J. A. De Loera and C. Meroni: The Best Ways to Slice a Polytope, Mathematics of Computations (2024)

work page 2024

[6] [6]

Breiding, B

P. Breiding, B. Sturmfels, and K. Wang: Computing Arrangements of Hypersur- faces, arXiv:2409.09622

arXiv

[7] [7]

Dalbec, and B Sturmfels: Introduction to Chow forms , Invariant Methods in Discrete and Computational Geometry: Proceedings of the Curac ¸ao Conference, 13–17 June, 1994 (1995)

J. Dalbec, and B Sturmfels: Introduction to Chow forms , Invariant Methods in Discrete and Computational Geometry: Proceedings of the Curac ¸ao Conference, 13–17 June, 1994 (1995)

work page 1995

[8] [8]

Dian, and P

G. Dian, and P. Heslop: Amplituhedron-like geometries, Journal of High Energy Physics No. 11 (2021)

work page 2021