Pith. sign in

REVIEW 3 major objections 3 minor 43 references

Evasive sets, twisted varieties, and container-clique trees

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that optimal-size evasive sets exist with thresholds r of size O(n^{1/k+...+1}) and are asymptotically best possible among complete intersections, and that there are at most 2^{O(q^{n-k})} such sets.

desk verdict A strong paper with a genuinely new container technique and plausible main theorems; the main gap is the sketched Claim 3.1, which is load-bearing for Theorems 1.1–1.2. read the letter →

arxiv 2507.07594 v1 pith:4SZCQLCU submitted 2025-07-10 math.CO

Jeck Lim , Jiaxi Nie , Ji Zeng This is my paper
classification math.CO MSC 05D4014G1514M10
keywords evasivesetstwistedvarietiescompleteintersectionscontainermethodcontainer-cliquetreesfinitefieldsgeneralpositioncollineartriples
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 optimal-size evasive sets can be built with much smaller intersection thresholds than previously known. Specifically, for fixed $d$ and $k$, it proves the existence of a $(d,k,r)$-evasive set in $\mathbb{F}_q^n$ of size $\Omega(q^{n-k})$ with $r = c_{d,k}\, n^{1/k + \cdots + 1/2 + 1}$, whereas earlier constructions needed $r$ polynomial in $n$ with exponent at least $k$. The engine is a new result in algebraic geometry: in $\mathbb{P}^n$ over an algebraically closed field there is a $d$-twisted complete intersection of dimension $n-k$ with degree $O(n^{1/k + \cdots + 1})$, and this degree is asymptotically tight among complete intersections for fixed $d,k$. The same paper counts evasive sets, showing there are at most $2^{O(q^{n-k})}$ $(k,r)$-evasive sets in $\mathbb{F}_q^n$, via a new variant of the container method called container-clique trees. A sympathetic reader would care because evasive sets underlie applications in Ramsey theory, incidence geometry, and error-correcting codes, and the smaller the threshold $r$ the stronger those applications become.

What carries the argument

A $d$-twisted variety: a variety $V$ in $\mathbb{P}^n$ whose intersection with every variety of codimension $\dim(V)$ and degree at most $d$ has dimension zero. The existence proof is a dimension count on the parameter space of polynomial tuples $(f_1,\ldots,f_k)$ of degrees $d_i \approx n^{1/(k+1-i)}$. For each $k$-cycle $X$, the bad set $B_X$ of tuples whose zero locus meets $|X|$ in positive dimension is bounded in codimension by Claim 3.1 using the Hilbert-function lower bound $\varphi_V(d) \ge \binom{d+k}{k}$ and the dimension formula for Chow varieties; the total bad locus is then lower-dimensional, so a good tuple exists. The enumeration machinery is the container-clique tree: a rooted tree whose nodes carry a shrinking container $C$ and a list of deleted large cliques, allowing the hypergraph container lemma to be applied only after rich $k$-flats have been removed. Because independent sets meet cliques in fewer than $r$ points, the final count is controlled by the number of leaves, at most $2^{O(q^{n-k})}$.

What would settle it

Compute, for a minimal nontrivial case such as $k=2$, $\ell=2$ in $\mathbb{P}^3$ and a fixed 2-cycle $X$, the dimension of the set $B_X$ of polynomial pairs $(f_1,f_2)$ with $\dim(Z(f_1,f_2)\cap |X|)>0$, and compare it with the claimed codimension $\min\{\binom{d_1+2}{2},\binom{d_2+1}{1}\}$. The paper explicitly skips the argument that $B_X$ is a Zariski-closed subvariety of the asserted codimension, so exhibiting an $X$ where the codimension is smaller, or where $B_X$ fails to be closed, would break the dimension count that produces the twisted variety.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: for every degree bound $d$ and codimension $k$, and over any algebraically closed field $\mathbb{F}$, there exists a $d$-twisted complete intersection variety $V \subset \mathbb{P}^n$ of dimension $n-k$ whose degree is at most $c_{d,k}\, n^{1/k + 1/(k-1) + \cdots + 1}$. The word 'twisted' means that $V$ intersects any variety of complementary dimension and degree at most $d$ in a zero-dimensional set, which is the projective-geometric version of evasiveness. The paper shows that for fixed $d,k$ this degree bound cannot be improved asymptotically among complete intersections: any $d$-twisted complete intersection must have degree $\Omega(n^{1/k + \cdots + 1})$, by a criterion on containing $k$-planes. In finite fields, the same construction, combined with the standard point-counting bound and an intersection degree count, yields a $(d,k,r)$-evasive set of size $(1\pm o(1))q^{n-k}$ with $r$ of the stated size. Theorem 1.3 then bounds the total number of $(k,r)$-evasive sets by $2^{O(q^{n-k})}$, and the container-clique tree technique used there also gives a streamlined proof of the known three-regime characterization of collinear-triple-free subsets of a random subset of $\mathbb{F}_q^2$.

Load-bearing premise

The load-bearing premise is Claim 3.1: for every $k$-cycle $X$, the set $B_X$ of polynomial tuples whose common zero locus meets $|X|$ in dimension greater than $k-\ell$ is declared to be a subvariety of codimension at least $\min_i \binom{d_i+k+1-i}{k+1-i}$; the paper states that the variety argument is the same as for $B$ and omits it, so the dimension count that guarantees a good tuple exists collapses if this codimension is any smaller.

Editorial extensions

If this is right

  • For every fixed $d,k$, random algebraic varieties defined by polynomials of degrees $n^{1/i}$ for $1\le i\le k$ give $(d,k,r)$-evasive sets of size $\Omega(q^{n-k})$ with $r = O(n^{1/k+\cdots+1})$, a polynomial improvement in $n$ over prior constructions.
  • The degree bound for $d$-twisted complete intersections is optimal up to a constant depending on $d,k$, so any further improvement would require varieties that are not complete intersections.
  • There are at most $2^{O(q^{n-k})}$ $(k,r)$-evasive sets in $\mathbb{F}_q^n$, matching the trivial $2^{\Omega(q^{n-k})}$ lower bound up to constants in the exponent.
  • The container-clique tree method yields the upper bound $\alpha(\mathbb{F}_q^2,p) \le (1+o(1))pq$ in the dense random regime and a simpler proof of the full three-regime characterization.
  • The same method improves the count of general position sets in $\mathbb{F}_q^n$ to $2^{q + q^{2/3}+o(1)}$.

Reading between the lines

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

  • If non-complete-intersection twisted varieties exist with degree $o(n^{1/k+\cdots+1})$, the same translation would produce evasive sets with even smaller $r$; the paper itself leaves this geometric possibility open.
  • Container-clique trees are a general container-method variant: any enumeration problem whose supersaturation can be proved only after deleting rich cliques should be amenable to the same leaf-and-label counting, with counting arcs or $H$-free hypergraphs as natural candidates.
  • The construction is a randomized algorithm by the polynomial identity lemma; whether the good tuple can be found deterministically with comparable $r$ is not addressed, and a positive answer would make these evasive sets useful in explicit algorithmic settings.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies (d,k,r)-evasive sets in F_q^n. The main existence theorem (Theorem 1.1) asserts the existence of such sets of size Ω(q^{n-k}) with r = O(n^{1/k+...+1}) for fixed d,k, improving earlier bounds. This is derived from Theorem 1.2, a dimension-counting construction of d-twisted complete intersection varieties in P^n of dimension n-k and degree O(n^{1/k+...+1}), together with a matching lower bound for complete intersections. The second main result, Theorem 1.3, gives an upper bound 2^{O(q^{n-k})} on the number of (k,r)-evasive sets, proved by a new 'container-clique tree' variant of the hypergraph container method. The method is also used to give a short proof of a random-Turán result for collinear-triple-free subsets of F_q^2.

Significance. If the proofs are completed, the results are significant: the evasive-set construction achieves optimal size with much smaller r than previous constructions, and the enumerative bound matches the trivial lower bound up to the exponent. The container-clique tree technique appears genuinely new and may have further applications. The paper also gives a cleaner proof of the Chen–Liu–Nie–Zeng random collinear-triple-free result with a sharp (1±o(1)) constant. The arguments are mostly self-contained and use standard algebraic geometry; the paper is refreshingly free of parameter fitting or circularity. However, as discussed below, two load-bearing lemmas in the algebraic-geometry section and one inequality in the container argument require repair before the main theorems are fully established.

major comments (3)
  1. [Section 3, Claim 3.1] Claim 3.1 is stated only for cycles X ∈ Ch(d,k,n), i.e., for k-dimensional varieties, but the inductive step 'apply inductive hypothesis to the variety Z(f1) ∩ |X|' requires the claim for (k−1)-dimensional varieties. This is outside the stated hypothesis; as written, the induction is invalid. Moreover, the subvariety assertion for B_X is explicitly skipped. Since inequality (3.1) and the dimension count after (3.2) depend on Claim 3.1, Theorems 1.1 and 1.2 are not fully supported as written. The claim is plausible and appears repairable by generalizing it to arbitrary varieties of dimension at most k, with the same formal codimension bound (for a variety W of dimension m, the bound is min_i binom(d_i+m+1-i, m+1-i), which for m = k−1 matches the stated min over i≥2), but the generalization and its proof must be supplied.
  2. [Section 3, 'moreover' part of Theorem 1.2] The proof that a d-twisted complete intersection V has degree Ω(n^{1/k+...+1}) contains a logical error: from a (k+1−i)-plane F contained in Z(f1,...,fi), the text takes a hyperplane H containing F and notes H∩V contains a positive-dimensional set, claiming this contradicts 1-twistedness. But H is not a k-dimensional variety, and a positive-dimensional intersection with a hyperplane does not by itself give a k-plane intersecting V in positive dimension. The contradiction is obtained instead by taking a k-plane F' containing F; then V∩F' contains F∩Z(f_{i+1},...,f_k), a positive-dimensional set. The proof should be corrected accordingly.
  3. [Section 5, proof of Theorem 1.3] The inequality chain '(|C|−|C'|)·θ|E|/|V| ≥ ... ≥ |E(H'[C\C'])| ≥ c|E|' is not justified: Lemma 4.2(c) only gives |E(H'[C'])| ≤ (1−c)|E|, which says nothing about the number of edges entirely in C\C'. Consequently the conclusion |C'| ≤ (1−c/θ)|C|, which is used to bound the height of the container-clique tree by O(log q), does not follow as written. The intended bound can be recovered by observing that every edge not contained in C' has at least one vertex in C\C', so the sum of degrees over C\C' is at least c|E|; this yields the same inequality with |E(H'[C\C'])| replaced by the number of edges intersecting C\C'. The proof should be amended.
minor comments (3)
  1. [Section 5, final counting display] The expression 'binom(q^2, r)^{√q log q}' should use κ = q^k rather than q^2, since cliques in this setting are subsets of k-flats in F_q^n and have size at most q^k. The final bound 2^{O(q^{n-k})} is unaffected.
  2. [Section 5, proof of Lemma 5.1, first case] The conclusion 'Pr[dim span(S) ≤ k] > 1−o(1)' is weaker than what the union bound actually gives (dim span(S) ≤ k−2). The weaker statement is sufficient for the subsequent counting, but stating the stronger conclusion would clarify the argument.
  3. [Section 3, Fact 2.1] The proof that a dense subset of tuples defines a reduced subscheme is very terse; citing a standard reference for the fact that the condition of being a complete intersection with the Jacobian criterion is open would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the evasive-set construction is based on an explicit dimension count with external algebraic-geometry inputs, and the one flagged issue is a proof gap rather than a circular reduction.

full rationale

The central derivation is self-contained and does not reduce to its inputs. Theorem 1.2 is proved by counting the bad tuples (f_1,...,f_k) whose zero locus fails twistedness, and the polynomial degrees d_i are chosen afterward to make the codimension estimate exceed dim Ch(d,k,n); the asserted degree bound is a consequence of this choice, not a fitted target. The dimension-counting ingredients, Lemma 2.2, Theorem 2.4, and the lower-bound tool Theorem 2.3 (Debarre--Manivel), are external results with hypotheses that do not include the conclusion. Theorem 1.1 is obtained by taking F_q-points and applying the external Lang--Weil and Bézout estimates, so its size bound is not assumed. The enumeration Theorem 1.3 is proved through the container method, independently of the existence theorem. The re-proof of the authors' earlier Theorem 1.4 is explicitly an illustration of the new container-clique technique and is not load-bearing for Theorems 1.1--1.3; even the parts referring to [10] do not assume Theorem 1.4 as an input, since the paper supplies a new proof of its essential third-range statement. The skeptic's concern about Claim 3.1 is a genuine proof gap: the induction appears to apply the claim to arbitrary subvarieties outside its stated hypothesis and the variety argument for B_X is skipped. But that is an incompleteness in proving a needed lemma, not a circularity in which a prediction is forced by definition or by a self-citation chain. No fitted parameter is renamed a prediction, and no load-bearing conclusion is imported from the authors' prior work.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claims rest on standard algebraic geometry and hypergraph container results. No free parameters are fitted to data; the polynomial degrees are chosen to satisfy explicit inequalities. The only new object, container-clique trees, is a proof device with no independent physical or geometric content, so no invented entities are listed.

assumptions (9)
  • standard math Fact 2.1: a dense set of polynomial tuples defines a reduced complete intersection.
    Used in the proof of Theorem 1.2 to pass from avoiding bad cycles to a reduced complete intersection variety.
  • standard math Lemma 2.2: Hilbert function lower bound phi_V(d) >= binom(d+k,k).
    Provides the codimension lower bound in Claim 3.1, which drives the dimension count.
  • standard math Theorem 2.3 (Debarre-Manivel): plane-containing criterion for complete intersections.
    Used to prove the asymptotic lower bound on degrees in Theorem 1.2.
  • standard math Theorem 2.4: dimension of the Chow variety is the stated maximum, which is O(n).
    Used to choose polynomial degrees large enough to dominate the bad locus.
  • standard math Lemma 2.6: fiber-dimension semicontinuity for projective morphisms.
    Used to show that the bad set B is a variety.
  • standard math Lang-Weil bound for F_q-points on varieties.
    Used in Theorem 1.1 to guarantee (1-o(1))q^(n-k) rational points on the twisted variety.
  • standard math Schwartz-Zippel lemma over finite fields.
    Used to show that most polynomial tuples over F_q are good.
  • standard math Hypergraph container lemma (Lemma 4.2, from Balogh-Morris-Samotij and Saxton-Thomason).
    Core tool in Theorems 1.3 and 1.4.
  • standard math Fact 5.2: a linear-algebra exchange fact about affine independence.
    Used in Lemma 5.1 to count edges in the random construction.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Evasive sets, twisted varieties, and container-clique trees." pith.science (2026). https://pith.science/paper/4SZCQLCU

@misc{pith2026250707594,
  author       = {Pith},
  title        = {Pith review of: Evasive sets, twisted varieties, and container-clique trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4SZCQLCU}},
  note         = {Machine review of arXiv:2507.07594}
}
abstract

In the affine space $\mathbb{F}_q^n$ over the finite field of order $q$, a point set $S$ is said to be $(d,k,r)$-evasive if the intersection between $S$ and any variety, of dimension $k$ and degree at most $d$, has cardinality less than $r$. As $q$ tends to infinity, the size of a $(d,k,r)$-evasive set in $\mathbb{F}_q^n$ is at most $O\left(q^{n-k}\right)$ by a simple averaging argument. We exhibit the existence of such evasive sets of sizes at least $\Omega\left(q^{n-k}\right)$ for much smaller values of $r$ than previously known constructions, and establish an enumerative upper bound $2^{O(q^{n-k})}$ for the total number of such evasive sets. The existence result is based on our study of twisted varieties. In the projective space $\mathbb{P}^n$ over an algebraically closed field, a variety $V$ is said to be $d$-twisted if the intersection between $V$ and any variety, of dimension $n - \dim(V)$ and degree at most $d$, has dimension zero. We prove an upper bound on the smallest possible degree of twisted varieties which is best possible in a mild sense. The enumeration result includes a new technique for the container method which we believe is of independent interest. To illustrate the potential of this technique, we give a simpler proof of a result by Chen--Liu--Nie--Zeng that characterizes the maximum size of a collinear-triple-free subset in a random sampling of $ \mathbb{F}_q^2$ up to polylogarithmic factors.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

43 extracted references · 40 canonical work pages

  1. [1]

    P. Azcue. On the dimension of the Chow varieties . Harvard University, 1992. PhD Dissertation

    work page 1992

  2. [2]

    Balogh and H

    J. Balogh and H. Luo. Maximum number of points in general position in a random subset of finite 3-dimensional spaces. arXiv preprint arXiv:2503.04102 , 2025

    work page arXiv 2025

  3. [3]

    Balogh, R

    J. Balogh, R. Morris, and W. Samotij. Independent sets in hypergraphs. Journal of the American Mathematical Society, 28(3):669–709, 2015

    work page 2015

  4. [4]

    Balogh and W

    J. Balogh and W. Samotij. The number of Ks,t-free graphs. Journal of the London Mathematical Society , 83(2):368–388, 2011

    work page 2011

  5. [5]

    Balogh and J

    J. Balogh and J. Solymosi. On the number of points in general position in the plane. Discrete Analysis, 16:1–20, 2018

    work page 2018

  6. [6]

    Bhowmick and O

    K. Bhowmick and O. Roche-Newton. Counting Arcs in F2 q. Discrete & Computational Geometry , 72(4):1630– 1646, 2024

    work page 2024

  7. [7]

    P. V. Blagojevi´ c, B. Bukh, and R. Karasev. Tur´ an numbers forKs,t-free graphs: topological obstructions and algebraic constructions. Israel Journal of Mathematics , 197(1):199–214, 2013

    work page 2013

  8. [8]

    Bowtell and P

    C. Bowtell and P. Keevash. The n-queens problem. arXiv preprint arXiv:2109.08083 , 2021

    arXiv 2021

Show all 43 references
  1. [9]

    B. Bukh. Extremal graphs without exponentially small bicliques. Duke Mathematical Journal , 173(11):2039– 2062, 2024

    2024

  2. [10]

    Y. Chen, X. Liu, J. Nie, and J. Zeng. Random Tur´ an and counting results for general position sets over finite fields. Science China Mathematics , to appear

  3. [11]

    Chow and B

    W.-L. Chow and B. L. van der Waerden. Zur algebraischen Geometrie. IX: ¨Uber zugeordnete Formen und algebraische Systeme von algebraischen Mannigfaltigkeiten. Mathematische Annalen, 113(1):692–704, 1937

    1937

  4. [12]

    D. Conlon. Private communication

  5. [13]

    Debarre and L

    O. Debarre and L. Manivel. Sur la vari´ et´ e des espaces lin´ eaires contenus dans une intersection compl` ete. Mathematische Annalen, 312:549–574, 1998. 14

    1998

  6. [14]

    R. A. DeMillo and R. J. Lipton. A Probabilistic Remark on Algebraic Program Testing. Information Processing Letters, 7(4):193–195, 1978

    1978

  7. [15]

    Z. Dvir, J. Koll´ ar, and S. Lovett. Variety evasive sets. computational complexity, 23:509–529, 2014

    2014

  8. [16]

    Dvir and S

    Z. Dvir and S. Lovett. Subspace evasive sets. In Proceedings of the forty-fourth annual ACM Symposium on Theory of Computing , pages 351–358, 2012

    2012

  9. [17]

    Erd¨ os, P

    P. Erd¨ os, P. Frankl, and V. R¨ odl. The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs and Combinatorics , 2(1):113–121, 1986

    1986

  10. [18]

    Erdos, D

    P. Erdos, D. J. Kleitman, and B. L. Rothschild. Asymptotic enumeration of Kn-free graphs. Colloquio Inter- nazionale sulle Teorie Combinatorie (Rome, 1973) , 2(17):19–27, 1976

    1973

  11. [19]

    Ferber, G

    A. Ferber, G. McKinley, and W. Samotij. Supersaturated sparse graphs and hypergraphs. International Math- ematics Research Notices, 2020(2):378–402, 2020

    2020

  12. [20]

    F¨ uredi, A

    Z. F¨ uredi, A. Naor, and J. Verstra¨ ete. On the Tur´ an number for the hexagon. Advances in Mathematics , 203(2):476–496, 2006

    2006

  13. [21]

    I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, Resultants, and Multidimensional Deter- minants. Birkh¨ auser, Boston, MA, 1994

    1994

  14. [22]

    Griffiths and J

    P. Griffiths and J. Harris. Principles of Algebraic Geometry . John Wiley & Sons, 1978

    1978

  15. [23]

    Z. Guo. Variety evasive subspace families. computational complexity, 33(2):10, 2024

    2024

  16. [24]

    Guruswami

    V. Guruswami. Linear-algebraic list decoding of folded Reed-Solomon codes. In 2011 IEEE 26th Annual Conference on Computational Complexity , pages 77–85. IEEE, 2011

    2011

  17. [25]

    Hartshorne

    R. Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics. Springer, 1977

    1977

  18. [26]

    Ihringer and J

    F. Ihringer and J. Verstra¨ ete. Large ( k; r, s; n, q)-sets in Projective Spaces. arXiv preprint arXiv:2211.04329 , 2022

    2022 arXiv

  19. [27]

    P. Keevash. Counting designs. Journal of the European Mathematical Society , 20(4):903–927, 2018

    2018

  20. [28]

    D. J. Kleitman and K. J. Winston. On the number of graphs without 4-cycles. Discrete Mathematics, 41(2):167– 172, 1982

    1982

  21. [29]

    Lang and A

    S. Lang and A. Weil. Number of points of varieties in finite fields. American Journal of Mathematics , 76(4):819– 827, 1954

    1954

  22. [30]

    B. Lehmann. Asymptotic behavior of the dimension of the Chow variety. Advances in Mathematics, 308:815–835, 2017

    2017

  23. [31]

    Milojevi´ c, I

    A. Milojevi´ c, I. Tomon, and B. Sudakov. Incidence bounds via extremal graph theory. arXiv preprint arXiv:2401.06670, 2024

    2024 arXiv

  24. [32]

    Morris and D

    R. Morris and D. Saxton. The number of C2ℓ-free graphs. Advances in Mathematics , 298:534–580, 2016

    2016

  25. [33]

    Nagle, V

    B. Nagle, V. R¨ odl, and M. Schacht. Extremal hypergraph problems and the regularity method. In Topics in Discrete Mathematics: Dedicated to Jarik Neˇ setˇ ril on the Occasion of his 60th Birthday, pages 247–278. Springer, 2006

    2006

  26. [34]

    R. Nenadov. The number of arcs in F2 q of a given cardinality. Discrete & Computational Geometry , to appear

  27. [35]

    Pudl´ ak and V

    P. Pudl´ ak and V. R¨ odl. Pseudorandom sets and explicit construction of Ramsey graphs.Quaderni di Matematica, 13:327––346, 2004

    2004

  28. [36]

    Roche-Newton and A

    O. Roche-Newton and A. Warren. Arcs in F2 q. European Journal of Combinatorics , 103(103512):1–15, 2022

    2022

  29. [37]

    P. Samuel. M´ ethodes d’Alg` ebre Abstraite en G´ eom´ etrie Alg´ ebrique. Springer, corrected 2nd edition, 1967

    1967

  30. [38]

    Saxton and A

    D. Saxton and A. Thomason. Hypergraph containers. Inventiones Mathematicae, 201(3):925–992, 2015

    2015

  31. [39]

    J. T. Schwartz. Fast Probabilistic Algorithms for Verification of Polynomial Identities. Journal of the ACM , 27(4):701–717, 1980

    1980

  32. [40]

    Simkin and Z

    M. Simkin and Z. Luria. A lower bound for the n-queens problem. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms , pages 2185–2197. SIAM, 2022

    2022

  33. [41]

    M. Sombra. Bounds for the Hilbert function of polynomial ideals and for the degrees in the Nullstellensatz. Journal of Pure and Applied Algebra , 117:565–599, 1997

    1997

  34. [42]

    Sudakov and I

    B. Sudakov and I. Tomon. Evasive sets, covering by subspaces, and point-hyperplane incidences. Discrete & Computational Geometry, 72(3):1333–1347, 2024

    2024

  35. [43]

    R. Zippel. Probabilistic Algorithms for Sparse Polynomials. In Symbolic and Algebraic Computation , volume 72 of Lecture Notes in Computer Science , pages 216–226. Springer, 1979. 15

    1979

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.