Two-Cut Coherence of Quintic Forms: Lifting Separations and Second-Derivative Completeness

Karthik Sheshadri

Two-Cut Coherence of Quintic Forms: Lifting Separations and Second-Derivative Completeness

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Pith's one-line read This paper introduces a two-cut coherence measure and proves that for quintic forms it is controlled by the largest second-derivative slice rank.

desk verdict A genuine new invariant with a sound quintic completeness theorem; the proofs hold up and the paper is worth refereeing, though presentation needs cleanup. read the letter →

arxiv 2608.07611 v1 pith:SODNJ35L submitted 2026-08-06 cs.CC

Karthik Sheshadri This is my paper · ORCID

classification cs.CC MSC 68Q1715A69

keywords two-cutcoherencerestrictedstrengthpolynomialslicerankbordercomplexityalgebraicbranchingprogramstensor-trainranksquinticformssecond-derivativecompleteness

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

Two-cut coherence $C_{k,\ell}(f)$ asks for the smallest number $r$ such that a homogeneous polynomial $f$ can be written as $\sum_{i,j} p_i m_{ij} q_j$ with the two degree interfaces $k$ and $\ell$ realized by one shared factorization. The paper's main theorem says that for quintics at cuts $(1,3)$, this joint cost is not a hidden new hardness: when the degree-$3$ restricted strength $C_3(f)$ is bounded, $C_{1,3}(f)$ and its border version are equivalent up to constant factors to $\Delta_2(f)$, the largest polynomial slice rank of a square second directional derivative of $f$. The paper also gives explicit quintics, built from Fermat cubics, whose individual cuts each have ordinary and border value $2$ but whose two-cut coherence grows linearly, so the joint cost can genuinely exceed the sum of local costs.

What carries the argument

The load-bearing identity is the three-group Leibniz rule. If $H=p^\top M q$ is a width-$r$ representation with linear entries in $p$ and quadratic entries in $M,q$, then every square second derivative $\partial_u^2 H$ expands into three grouped sums, each of polynomial slice rank at most $r$, giving $\Delta_2(H)\le 3C_{1,3}(H)$. The completeness direction uses the fact that point-evaluation functionals $Q\mapsto Q(u)$ span the dual of any finite-dimensional space of quadrics over an infinite field, so square second derivatives of quadrics act as a dual basis. Choosing directions $u_j$ dual to the quadratic endpoint space of a minimal $C_3$-decomposition isolates each hidden cubic coefficient up to a Leibniz error of slice rank at most $2t$, and slicing those cubics gives the bound $t\Delta_2(f)+2t^2$.

What would settle it

For the explicit family $F_n=ab\sum_{i=1}^n v_i^3+cd\,v_1^3$ over $\mathbb{C}$, compute $C_{1,3}(F_n)$ and compare with $t\Delta_2(F_n)+2t^2$ where $t=2$; the theorem forces $\Omega(n)\le C_{1,3}(F_n)\le 2\Delta_2(F_n)+8$, so a direct computation exceeding that upper band would refute Theorem 1.1. In the other direction, a bounded-$C_3$ quintic family with $\Delta_2=O(1)$ and $C_{1,3}\to\infty$ would refute the completeness claim.

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Extended reading notes

Core claim

Over an algebraically closed field of characteristic zero, every nonzero quintic $f$ with $t=C_3(f)$ and $\Delta_2(f)=\max_u C_1(\partial_u^2 f)$ satisfies $\lceil \Delta_2(f)/3\rceil \le \overline{C}_{1,3}(f)\le C_{1,3}(f)\le t\Delta_2(f)+2t^2$. Hence at bounded $C_3$, ordinary two-cut coherence, border two-cut coherence, and the one-cut obstruction read from second derivatives all grow together up to constants. The companion separation result is that for coprime nonzero cubics $A,B$, the quintic $L=abA+cdB$ has $C_1(L)=C_3(L)=2$ (ordinary and border), yet $\lceil \max\{C_1(A),C_1(B)\}/3\rceil \le \overline{C}_{1,3}(L)\le C_1(A)+C_1(B)$; with $A$ the Fermat cubic $\sum_i v_i^3$, whose slice rank is $\lceil n/2\rceil$, the gap is unbounded and border-stable, refuting any universal bound $C_{k,\ell}\le C_k+C_\ell$.

Load-bearing premise

The upper-bound proof must choose derivative directions that separate the quadratic factors of a minimal degree-three decomposition, and that pointwise separation of an arbitrary finite-dimensional space of quadrics requires an infinite ground field; over a finite field the isolation step and the bound $t\Delta_2(f)+2t^2$ would fail.

Editorial extensions

If this is right

For fixed $t_0$, every quintic with $C_3(f)\le t_0$ satisfies $\Delta_2(f)/3\le C_{1,3}(f)\le \overline{C}_{1,3}(f)\le (t_0+2t_0^2)\Delta_2(f)$, so at bounded degree-three strength the two-cut complexity cannot escape second-derivative detection.
Two-cut coherence is exactly the minimum interface width of a three-block compressed transfer network and the minimum, over all tensor lifts of $f$ through commutative multiplication, of the larger of the two tensor-train endpoint ranks; since every homogeneous ABP yields such a lift, $C_{k,\ell}$ lower-bounds homogeneous ABP width.
The Fermat-family quintics $F_n=ab\sum_{i=1}^n v_i^3+cd\,v_1^3$ have $C_1=C_3=2$ in both ordinary and border senses but two-cut coherence at least $\lceil \lceil n/2\rceil/3\rceil$, giving an explicit $\Omega(n)$ homogeneous-ABP width lower bound for a polynomial with an $O(n)$-size formula.
No family of quintics satisfies $C_3(f_n)=O(1)$, $\overline{C}_{1,3}(f_n)\to\infty$, and $\Delta_2(f_n)=o(\overline{C}_{1,3}(f_n))$; the completeness inequality rules out such extraction-invisible families.
The universal additive bound $C_{k,\ell}\le C_k+C_\ell$ is false, since for $n\ge 25$ the Fermat construction gives $C_{1,3}(F_n)>C_1(F_n)+C_3(F_n)$.

Reading between the lines

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

If a similar completeness theorem held for other cut profiles, restricted-strength lower bounds in those regimes would reduce to slice-rank computations on derivatives; the quintic proof suggests the key feature is that quadrics are dual to square second-order evaluations, so a generalization would need a new source of dual bases for the relevant endpoint spaces.
The Fermat construction currently yields only a linear ABP-width lower bound because the slice rank of a Fermat cubic is linear in the number of variables; replacing it with a family of cubics of larger slice rank would immediately magnify the separation between local and common interface widths.
The multiplication-fiber formulation invites a concrete test: numerically optimize tensor-train endpoint ranks over the affine fiber $\mu^{-1}(f)$ for random quintics and check how tightly $\Delta_2(f)$ tracks the resulting width; the theorem implies the ratio stays bounded whenever $C_3(f)$ is bounded.
Because the sandwich equates border and ordinary two-cut coherence up to constants for quintics, any hypothetical border-ABP width gap for these families would have to originate outside the two-cut interface, for instance in the hidden quadratic transfer labels.

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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

The main news is that two-cut coherence is a genuinely new measure, and the main completeness theorem for quintics checks out. I went through the proof of Theorem 1.1: the three-group Leibniz identity gives the D/3 lower bound, and the point-evaluation argument in Lemma 6.1 supplies the tD+2t^2 upper bound. The lifting separation is also solid, and the Fermat cubic computation is simple and correct. This refutes any universal additive bound of the form C_{k,l} <= C_k + C_l, and the resulting linear lower bound on homogeneous ABP width for that easy family is a nice free consequence.

What the paper does well: it places the new invariant in an exact computational model (compressed transfer networks, multiplication-fiber tensor-train width), and it is honest about scope. It does not pretend to touch VP vs VNP; it only gives linear lower bounds for an easy family, and it says so explicitly. The proofs are checkable, with clean lemmas and a conservative novelty boundary.

The soft spots are presentation, not math. The delta-placement in Lemma 4.1 is confusing on first read, and the display around equation (3) in the abstract appears to reverse ordinary and border complexity relative to the body. These need fixing before publication, but they do not confuse the argument once the definitions are lined up. The supplementary verification scripts are referenced but not visible in the archived text; that is a minor concern because no main result depends on them. The 'border-stable' terminology could be aligned more carefully with existing border rank language, but the mathematical statement is clear.

I saw no significant objection; the stress-test note's concern about Lemma 6.1 requiring an infinite field is satisfied by the algebraically closed characteristic zero hypothesis, so the isolation argument is within the theorem's scope. The citation pattern is appropriate and the self-assessment of limitations is accurate.

This is a paper for people working on restricted strength, slice rank, and ABP width. It deserves a serious referee. My recommendation: send it to peer review with the expectation of minor revisions on presentation. The central claim is sound and the invariant deserves to be known.

Referee Report

0 major / 5 minor

Summary. The paper introduces a two-cut coherence parameter C_{k,l}(f), the least r such that a degree-d form f can be written as f = sum_{i,j=1}^r p_i m_{ij} q_j with deg p_i = k, deg m_{ij} = l-k, and deg q_j = d-l. It proves exact characterizations of this parameter in terms of compressed three-block transfer networks and as the minimum, over tensor lifts through commutative multiplication, of the larger endpoint tensor-train rank, and observes that C_{k,l} lower-bounds homogeneous ABP width. The main result, Theorem 1.1, is a second-derivative completeness theorem for quintics at cuts (1,3): with t = C_3(f) and D(f) = max_u C_1(partial_u^2 f), it establishes ceil(D(f)/3) <= Cbar_{1,3}(f) <= C_{1,3}(f) <= t D(f) + 2t^2 over an algebraically closed field of characteristic zero. The lower bound comes from a three-group Leibniz identity (Lemma 4.1) and the upper bound from a point-evaluation isolation argument (Lemma 6.1 and Theorem 6.2). The paper also proves a border-stable lifting separation for quintics of the form L = abA + cdB, and, using a computation of the restricted strength of the Fermat cubic (Proposition 5.2), obtains an unbounded gap between separately optimal local interfaces and a common interface. An appendix shows that the constant 3 in the Leibniz transfer is asymptotically sharp for arbitrary width-r representations, and supplementary exact verification scripts are provided.

Significance. If the results are correct, the paper introduces a genuinely new invariant that is finer than restricted strength and that has a clean computational interpretation: two-cut coherence is the minimum shared endpoint width of a three-block compressed transfer network, equivalently the minimum endpoint tensor-train rank over a commutative multiplication fiber. The main theorem is a dimension-free completeness result: for quintics with bounded C_3, the border and ordinary two-cut coherence are controlled up to constants by a single-cut obstruction exposed by second directional derivatives. This is a nontrivial and checkable structural statement. The lifting separation is also significant because it refutes any universal additive bound C_{k,l} <= C_k + C_l and does so in a border-stable way, so the gap is not an artifact of Zariski closure. The paper's strengths include explicit, checkable proofs, a clear statement of the exact boundary of the method, and exact computational verification scripts for the main polynomial identities; no lower bound depends on a numerical or heuristic computation.

minor comments (5)

[§4, Lemma 4.1] Equation (12) uses a compressed notation in which delta appears both as a derivative operator and as a subscript on q; as printed, the right-hand side appears to omit the terms (delta p)^T (delta M) q and (delta p)^T M (delta q) that arise when differentiating the linear factor vector p. Please rewrite the identity with explicit parentheses, for example 2(delta p)^T(delta M)q + p^T(delta^2 M)q + 2(delta p)^T M(delta q) + 2p^T(delta M)(delta q) + p^T M(delta^2 q), or add a sentence explaining that delta acts on p as well as on M and q. The grouping argument in the proof is correct once this notation is clarified.
[§3.3, Proposition 3.5] The proof of the upper bound w_ABP(f) <= N r^2 is under-specified: after introducing layer-two nodes indexed by (i,j,s), the text does not say how the two linear factors of each quadratic q_j are inserted, and a standard consecutive-layer homogeneous ABP appears to require an additional layer of nodes unless a nonstandard edge convention is being used. Please spell out the full layer-by-layer construction; the statement is plausible and this issue does not affect the main theorem, but as written the reader cannot verify the claimed width bound.
[§6.2, Eq. (22)] The normalization partial^2_{u_j} Q_i = delta_{ij} silently absorbs the factor 2 from the identity partial^2_u Q = 2Q(u); this is harmless, but a one-sentence reminder would prevent confusion when comparing with Lemma 6.1.
[Eqs. (3), (20), (18)] The ceiling brackets in the displayed inequalities are typeset as empty boxes in the arXiv rendering; please use \lceil and \rceil so that the statements involving ceil(D(f)/3) and the related bounds are unambiguous.
[§5.1, Proposition 5.2] In the reverse inequality of the Fermat-cubic proof, the sentence 'Polarizing the identity A_n|_W = 0 gives ...' is terse; writing out the substitution w + alpha u + beta v in W and taking the coefficient of alpha beta would make the derivation of sum_{i in S} w_i u_i v_i = 0 easier to verify, although the argument is sound.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified: the main bounds are derived from independent definitions and lemmas.

full rationale

The paper's central claim, Theorem 1.1, is a genuine sandwich inequality relating the newly defined two-cut coherence C_{1,3}(f) to the independently defined quantities C_3(f) and Delta_2(f). No equation in the derivation is equivalent to its own input by construction. The lower bound follows from the three-group Leibniz identity (Lemma 4.1), which is a direct product-rule computation and is independent of the conclusion. The upper bound, Theorem 6.2, chooses a minimal degree-3 decomposition, uses Lemma 6.1 to select directions whose square second derivatives evaluate the quadratic endpoint space, and isolates each cubic coefficient G_j with a controlled Leibniz error. This is a constructive extraction argument, not a restatement of the bound. The quantities C_3, Delta_2, and C_1 are all defined from the same polynomial but not from C_{1,3}, so there is no self-definitional circularity. The border-stable lifting theorem likewise uses explicit decompositions, coprimality, and the closedness of slice-rank loci, with no fitted parameters or prediction-after-fitting structure. The Fermat cubic slice-rank computation is self-contained. References to prior work are background and do not carry the proofs; no uniqueness theorem is imported from the authors, and no ansatz is smuggled in via citation. The paper explicitly does not claim ordinary one-cut complexity equals two-cut coherence, and Theorem 1.1's bounds are stated and proved with constants, not forced by normalization. Overall, the derivation chain is self-contained and no circular step was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim is carried by standard algebraic geometry and by the special duality of quadrics with point evaluations. No numeric parameters are fitted to make the theorem true; the quantities t and Delta_2 are defined by the polynomial, not adjusted. No new physical or mathematical entities are postulated.

assumptions (5)

domain assumption Ground field F is algebraically closed of characteristic zero and the polynomial ring R = Sym(V*) is standard graded.
Section 2 states this setting; it is needed for Zariski closures, polarization, and the derivative identities.
standard math Point evaluations e_u(Q) = Q(u) span the dual of every finite-dimensional space of quadrics over an infinite field (Lemma 6.1).
Theorem 6.2 uses this to choose directions u_j that isolate each cubic coefficient G_j in the double derivative.
standard math Images of projective varieties under regular maps are closed; slice-rank-at-most-s loci and rank-one product loci are closed.
Used in Proposition 2.3 and Section 5 to pass from ordinary to border lower bounds.
standard math Gauss's lemma and irreducibility in S[a] apply to L = abA + cdB when gcd(A,B) = 1.
Theorem 5.1 uses this to force local C_1 and C_3 values to be at least 2.
standard math Krull's height theorem gives a lower bound on the slice rank of disjoint monomials in Appendix A.
Appendix A uses this to prove the 3r-1 sharpness example for the three-group transfer.

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Cite this review

Pith. "Pith review of Two-Cut Coherence of Quintic Forms: Lifting Separations and Second-Derivative Completeness." pith.science (2026). https://pith.science/paper/SODNJ35L

@misc{pith2026260807611,
  author       = {Pith},
  title        = {Pith review of: Two-Cut Coherence of Quintic Forms: Lifting Separations and Second-Derivative Completeness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SODNJ35L}},
  note         = {Machine review of arXiv:2608.07611}
}

read the original abstract

For a homogeneous polynomial f of degree d, the degree-k restricted strength C_k(f) is the least number of products needed to write f with factor degrees k and d-k. We introduce a two-cut coherence parameter C_{k,l}(f): the least r such that f = sum_{i,j=1}^{r} p_i m_{ij} q_j with deg p_i = k, deg m_{ij} = l-k, and deg q_j = d-l. This requires two degree interfaces to be realized by a single common factorization. We show it equals the minimum common endpoint width of a three-block compressed transfer network, and equivalently the minimum, over all tensor lifts of f through commutative multiplication, of the larger of the two tensor-train endpoint ranks. In particular it lower-bounds homogeneous ABP width. Our main result is an extraction-completeness theorem for quintics at cuts (1,3). Let D(f) be the largest polynomial slice rank C_1 of a second directional derivative of f, and let t = C_3(f). Over an algebraically closed field of characteristic zero, ceil(D(f)/3) <= Cbar_{1,3}(f) <= C_{1,3}(f) <= t*D(f) + 2t^2, where Cbar denotes border complexity. Hence when C_3 is bounded, ordinary and border two-cut coherence are equivalent up to constants to a one-cut obstruction exposed by a second derivative. We also prove a border-stable lifting separation. For coprime nonzero cubics A and B, the quintic L = abA + cdB has ordinary and border local values C_1 = C_3 = 2, while ceil(max{C_1(A), C_1(B)}/3) <= Cbar_{1,3}(L) <= C_1(A) + C_1(B). Taking A to be a Fermat cubic in n variables, for which we show C_1 = ceil(n/2), gives an unbounded gap between separately optimal local interfaces and a common interface, even in border complexity. This refutes any universal bound of the form C_{k,l} <= C_k + C_l.

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Works this paper leans on

12 extracted references · 5 canonical work pages

[1]

Tigran Ananyan and Melvin Hochster. 2020. Small Subalgebras of Polynomial Rings and Stillman’s Conjecture.Journal of the American Mathematical Society33, 1 (2020), 291–309. doi:10.1090/jams/932

doi:10.1090/jams/932 2020
[2]

Edoardo Ballico and Emanuele Ventura. 2021. The Strength for Line Bundles.Math. Scand.127, 3 (2021), 405–440. doi:10.7146/math.scand.a-128529

work page doi:10.7146/math.scand.a-128529 2021
[3]

Eggermont

Arthur Bik, Jan Draisma, and Rob H. Eggermont. 2019. Polynomials and Tensors of Bounded Strength.Communications in Contemporary Mathematics 21, 7 (2019), 1850062. doi:10.1142/S0219199718500621

work page doi:10.1142/s0219199718500621 2019
[4]

Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, and Nitin Saurabh. 2020. Algebraic Branching Programs, Border Complexity, and Tangent Spaces. In35th Computational Complexity Conference (CCC 2020) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 169). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 21:1–21:24. doi:10.423...

doi:10.4230/lipics.ccc.2020.21 2020
[5]

Cornelius Brand, Petteri Kaski, and Jiaheng Wang. 2026. Partition Rank and Algebraic Circuit Lower Bounds. arXiv:2607.02241 [cs.CC] Accepted to ESA 2026; arXiv version 2

work page Pith review arXiv 2026
[6]

1995.Commutative Algebra with a View Toward Algebraic Geometry

David Eisenbud. 1995.Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, Vol. 150. Springer. doi:10.1007/978-1-4612-5350-1

doi:10.1007/978-1-4612-5350-1 1995
[7]

Cosimo Flavi, Fulvio Gesmundo, Alessandro Oneto, and Emanuele Ventura. 2025. Polynomials of Small Slice Rank and Strength. arXiv:2509.12322 [math.AG]

work page arXiv 2025
[8]

Fulvio Gesmundo, Purnata Ghosal, Christian Ikenmeyer, and Vladimir Lysikov. 2022. Degree-Restricted Strength Decompositions and Algebraic Branching Programs. In42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 250). Schloss Dags...

doi:10.4230/lipics 2022

Show all 12 references

[9]

W. T. Gowers. 2021. The Slice Rank of a Direct Sum. arXiv:2105.08394 [math.CO]

2021 arXiv
[10]

Noam Nisan. 1991. Lower Bounds for Non-Commutative Computation. InProceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing. Association for Computing Machinery, 410–418. doi:10.1145/103418.103462

1991
[11]

I. V. Oseledets. 2011. Tensor-Train Decomposition.SIAM Journal on Scientific Computing33, 5 (2011), 2295–2317. doi:10.1137/090752286

2011 doi
[12]

Ke Ye and Lek-Heng Lim. 2018. Tensor Network Ranks. arXiv:1801.02662 [math.NA] Manuscript submitted to ACM

2018 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Tigran Ananyan and Melvin Hochster. 2020. Small Subalgebras of Polynomial Rings and Stillman’s Conjecture.Journal of the American Mathematical Society33, 1 (2020), 291–309. doi:10.1090/jams/932

doi:10.1090/jams/932 2020

[2] [2]

Edoardo Ballico and Emanuele Ventura. 2021. The Strength for Line Bundles.Math. Scand.127, 3 (2021), 405–440. doi:10.7146/math.scand.a-128529

work page doi:10.7146/math.scand.a-128529 2021

[3] [3]

Eggermont

Arthur Bik, Jan Draisma, and Rob H. Eggermont. 2019. Polynomials and Tensors of Bounded Strength.Communications in Contemporary Mathematics 21, 7 (2019), 1850062. doi:10.1142/S0219199718500621

work page doi:10.1142/s0219199718500621 2019

[4] [4]

Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, and Nitin Saurabh. 2020. Algebraic Branching Programs, Border Complexity, and Tangent Spaces. In35th Computational Complexity Conference (CCC 2020) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 169). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 21:1–21:24. doi:10.423...

doi:10.4230/lipics.ccc.2020.21 2020

[5] [5]

Cornelius Brand, Petteri Kaski, and Jiaheng Wang. 2026. Partition Rank and Algebraic Circuit Lower Bounds. arXiv:2607.02241 [cs.CC] Accepted to ESA 2026; arXiv version 2

work page Pith review arXiv 2026

[6] [6]

1995.Commutative Algebra with a View Toward Algebraic Geometry

David Eisenbud. 1995.Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, Vol. 150. Springer. doi:10.1007/978-1-4612-5350-1

doi:10.1007/978-1-4612-5350-1 1995

[7] [7]

Cosimo Flavi, Fulvio Gesmundo, Alessandro Oneto, and Emanuele Ventura. 2025. Polynomials of Small Slice Rank and Strength. arXiv:2509.12322 [math.AG]

work page arXiv 2025

[8] [8]

Fulvio Gesmundo, Purnata Ghosal, Christian Ikenmeyer, and Vladimir Lysikov. 2022. Degree-Restricted Strength Decompositions and Algebraic Branching Programs. In42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 250). Schloss Dags...

doi:10.4230/lipics 2022