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Probabilistic Saturations and Alt's Problem

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Pith's one-line read The paper argues that a probabilistic finite-field computation giving 8,652 solutions outside the base locus completes Alt's problem by ruling out any four-bar linkages beyond the 4,326 already found numerically.

desk verdict A well-executed computational study whose advertised proof of certainty for Alt's problem collapses on a false finite-field claim. read the letter →

arxiv 1908.06020 v1 pith:G2JT7M7W submitted 2019-08-15 math.AG cs.SC

Jonathan D. Hauenstein , Martin Helmer This is my paper · ORCID
classification math.AGcs.SC MSC 14Q2013P1068W30
keywords Alt'sproblemfour-barlinkagesprobabilisticsaturationGröbnerbasesfinitefieldsHilbertfunctionsNewton-OkounkovbodiesChernclasses
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

Alt's 1923 problem asks how many four-bar linkages trace a coupler curve through nine prescribed points. A 1992 homotopy continuation computation found 8,652 isolated solutions, i.e. 4,326 distinct linkages, but that computation was numerical and could not rule out additional solutions. This paper tries to turn the count into a sharp upper bound by replacing the coefficients of the coupler-curve equations with independent random parameters and counting, outside the degenerate base locus $X$, the solutions to a system of general linear combinations over finite fields. The reported finite-field computations yield $g_0(X,\mathbb{C}^8)^{\mathrm{rand}}_{\mathbb{Z}_p}=8652$ for a range of primes, so the quotient dimension is an upper bound on the true complex degree; dividing by the relabeling symmetry gives at most 4,326 distinct four-bar linkages, and dividing by the full symmetry gives at most 1,442 coupler curves.

What carries the argument

The central object is the saturation count $g_i(X,\mathbb{C}^n)=\deg V(\Theta\cdot x-1,\Lambda\cdot f)\setminus X$: the number of solutions, counted away from the base locus $X$, of $i$ general affine linear equations in $x$ and $n-i$ general linear combinations of the defining polynomials. To compute it over $\mathbb{Z}_p$, the paper uses the ideal $I_i=\langle\Theta\cdot x-1,\Lambda\cdot f,1-(\mu\cdot f)T\rangle$, whose quotient dimension equals $g_i$ by Theorem 4.1 through the Rabinowitz trick. The probabilistic engine is Corollary 4.14, which states that with entries of $\Theta,\Lambda,\mu$ chosen uniformly from $\mathbb{Z}_p$, the probability that the finite-field quotient dimension equals the true complex value is at least $((p-1)/p)^\nu(1-(g_i+2n(D_{\min}+(r+n)D_{\max})\deg V)/p)$, with $\nu$ a Gröbner-basis size bound. This inequality is what converts a successful finite-field run into a probabilistic certificate and sets the required prime size.

What would settle it

Two observations would settle the matter. For the probability guarantee, test the proof's injectivity claim by evaluating a nonconstant leading-coefficient polynomial from Example 4.6 on the random set $\{0,\ldots,p-1\}$: a polynomial like $x^2-1$ vanishes on about $2/p$ of the set, so a leading coefficient with that behavior would invalidate the bound $((p-1)/p)^\nu$. For the numerical claim, compute $g_0(X,\mathbb{C}^8)$ over $\mathbb{Z}_p$ for a prime larger than $2^{116}$, or obtain a certified rational Gröbner basis for one random trial; any value other than 8652 would refute the claimed completion of Alt's problem.

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Extended reading notes

Core claim

The central claim is that Alt's problem has a complete answer: the number of distinct four-bar linkages whose coupler curve interpolates nine general points is 4,326, because the number of isolated solutions to the randomized saturation system, counted outside the base locus $X=V(f_1,\ldots,f_{15})$ of degenerate linkages, is exactly $g_0(X,\mathbb{C}^8)=8652$. The paper reports that, over the finite fields $\mathbb{Z}_p$ for the primes from $2^7+3$ through $2^{27}+29$, the dimension of the quotient $\mathbb{Z}_p[x,T]/I_0(\Theta,\Lambda,\mu)$ is always 8652, where $I_0$ is generated by eight general linear combinations of the fifteen $f_j$ together with the Rabinowitz-trick polynomial $1-(\mu\cdot f)T$. Since a zero-dimensional quotient over $\mathbb{Z}_p$ bounds the degree of the corresponding complex system when the prime is lucky, the paper concludes that no additional solutions can exist and that the upper bound $g_0/2=4326$ is sharp.

Load-bearing premise

The argument's weakest link is the claim, used in Theorem 4.7, that every nonconstant polynomial over $\mathbb{F}_p$ is injective and therefore vanishes on at most $1/p$ of the random choices; if that claim gives way, the lower bound $((p-1)/p)^\nu$ in Corollary 4.14 is not established.

Editorial extensions

If this is right

  • If $g_0(X,\mathbb{C}^8)=8652$ is accepted, Alt's problem is complete: exactly 4,326 distinct four-bar linkages and 1,442 distinct coupler curves pass through nine general points, because the known constructions meet the new upper bound.
  • The verified values $g_i(X,\mathbb{C}^8)$ for $i=1,\ldots,7$ certify the degrees of the varieties of linkages interpolating $9-i$ general points, upgrading the numerical table of [12] to probabilistic upper bounds for a family of synthesis problems.
  • The same saturation framework computes volumes of Newton-Okounkov bodies and characteristic classes such as Euler characteristics, Chern classes, and Segre classes, so the finite-field probability analysis gives explicit confidence levels for those computations.
  • Corollary 4.14 supplies an a priori recipe for prime size: for $g_0$, taking $p>2^{116}$ guarantees success probability exceeding 0.99, while the experiments show that much smaller primes already succeed in practice.
  • Because each finite-field run computes a zero-dimensional quotient, a successful run over a lucky prime yields an upper bound on the complex degree, not merely a numerical suggestion.

Reading between the lines

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

  • The numerical claim and the probability guarantee should be assessed separately: even if the injectivity assertion behind Theorem 4.7 is not valid, the empirical fact that many primes and floating-point trials return 8652 could still be true and could still make 4,326 the right answer.
  • A deterministic completion of Alt's problem would follow from one certified lucky-prime computation: find one parameter set whose Gröbner basis over $\mathbb{Z}_p$ is Pauer lucky, lift the computation to $\mathbb{Q}$, and the upper bound becomes unconditional; the present paper does not carry out that lift.
  • The coefficient-randomization trick is not specific to four-bar linkages: any enumerative problem whose base locus is known and whose symmetry group is understood can be bounded by the same $g_0$ computation, with the conics example in the paper serving as a small test case.
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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

3 major / 4 minor

Summary. The paper develops probabilistic saturation techniques for computing the numbers gi(X,C^n), which count solutions to a general affine-linear/linear-combination system outside the base locus X. These numbers arise in Alt's four-bar linkage problem, where g0(X,C^8)/2 is proposed as an upper bound on the number of distinct linkages interpolating nine general points. The manuscript combines certified numerical Hilbert function computations with symbolic upper bounds, then analyzes finite-field Gr"obner basis computations to give probabilistic guarantees. Section 5 reports modular computations matching the known degree g0(X,C^8)=8652, and Section 6 claims that verifying this value theoretically completes Alt's problem.

Significance. If the probabilistic certificate were sound, the paper would provide a practical and rigorous method for upper bounds in Alt's problem and for related projective degree, Segre class, Chern class, and Newton-Okounkov volume computations. The Hilbert-function methodology in Section 3, which certifies lower bounds with alphaCertified and matches them with symbolic upper bounds, is a genuine strength. The examples are detailed, and the reported timings and parameter choices make the experiments reproducible in principle. However, the central finite-field probability argument in Section 4 is invalid as written, and the advertised conclusion in Section 6 substantially overstates what the computations establish.

major comments (3)
  1. [§4.2, Theorem 4.7, Eq. (16)] The proof of Theorem 4.7 asserts that 'a nonconstant polynomial must map Z_p to itself injectively' and uses this to conclude that each leading coefficient is nonzero with probability at least (p-1)/p. This is false over finite fields: for example, f(t)=t^2-1 is nonconstant and vanishes at two elements of F_p. The error is load-bearing because the factor ((p-1)/p)^nu in (16) is the basis for Theorem 4.9 and Corollary 4.14. Moreover, the paper's own Example 4.6 gives leading coefficients such as c1(W)=(theta_denom1 theta_num2)^3 lambda_denom1 lambda_denom2 lambda_denom3 lambda_num4, which are products of several independently chosen variables, so P(c1 != 0)=((p-1)/p)^6, strictly smaller than (p-1)/p for every p>1. The cleared-denominator leading coefficients in I_i(W) have the same multiplicative structure. Therefore the claimed lower bound in (16) is not valid, and the field-size estimate p>2^116 in Section 5.1 is unsupported.
  2. [§4.1, Theorem 4.5] The proof of Theorem 4.5 contains a second unsupported claim. From a Zariski dense subset D on which p divides a leading coefficient c_j(W), the proof infers that c_j(W) must be constant because 'a nonconstant polynomial must map dense sets to dense sets' and the multiples of p are not dense in Q. This is false in the classical topology even over Q: t maps to t^2 sends Q to Q_{\ge 0}, which is not dense in Q. Furthermore, the desired conclusion does not follow: a nonzero polynomial all of whose coefficients are divisible by p would satisfy p | c_j(W) for every integer W without being constant. Since Theorem 4.7 refers back to the proof of Theorem 4.5 for the nonconstancy of the leading coefficients, the modular-luckiness framework is not established as written.
  3. [§6 and Corollary 4.14] The concluding statement that verifying g0(X,C^8)=8652 'theoretically completes Alt's problem by showing that there could be no additional solutions' is not supported by the manuscript. Section 5 reports finitely many successful random trials over selected finite fields and numerical homotopy runs; these are empirical confirmations. Even a corrected version of Corollary 4.14 would at most provide a high-confidence probabilistic statement rather than a theoretical proof. Unless a deterministic certificate is supplied, such as an exact Gr\"obner basis computation over Q or a certified interval/arithmetic argument, the conclusion should be rephrased as probabilistic confirmation, not theoretical completion.
minor comments (4)
  1. [§4.1, Example 4.6] The word 'Symoblically' should be 'Symbolically'.
  2. [§5.1, Corollary 5.1] The bound deg(V) <= 7,620,480,000 is stated without showing the B\'ezout calculation; please include the product of the degrees or the elimination argument used, since the reader cannot verify this number from the stated D_min and D_max alone.
  3. [Definition 4.4] The phrase 'p is larger than any coefficient' should specify that absolute values are meant, since coefficients may be negative.
  4. [Table 3] It would help to state explicitly which entries are in bold because they agree with Table 2, and to explain the missing entry for g0 at p=2^27+29.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: finite-field saturation values are computed independently and compared with benchmarks only after the fact.

full rationale

The claimed derivation is self-contained rather than circular. The quantities g_i(X,C^n) are defined in Eq. (2) by counting solutions outside X to a general linear combination of f, and Theorem 4.1 connects this to the dimension of R[T]/I_i(Theta,Lambda,mu); neither definition presupposes the 8652 count. The Alt upper-bound claim g_0(X,C^8)/2 follows by replacing the specialized coefficients c_j(p_i, bar p_i) with independent parameters, so it is a constructed bound, not a renamed numerical result. In Section 5 the values g_i(X,C^8)^{rand}_{Z_p} are produced by fresh modular Groebner basis computations; Table 3 even shows smaller primes giving values different from 8652 (e.g., 8492 for p=3), so the computation is not hard-wired to the benchmark. The agreement with [12, Table 1] is used only as post hoc confirmation. The authors' self-citations ([12,24,25]) are published background or benchmarks and are not load-bearing inputs forcing the conclusion. The paper's own Section 6 limitation — that Magma resources do not permit p>2^116, so the a priori probability bound cannot be directly realized — and the questionable injectivity assertion in Theorem 4.7 are correctness and confidence concerns, not circularity; they do not make the derivation equivalent to its inputs. No step reduces Eq. (10) to Eq. (2) by construction, and no fitted parameter is renamed as a prediction.

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

No fitted constants are introduced: the entries of Theta, Lambda, and mu are random inputs, and p is a user-chosen prime. The derivation relies on standard algebraic geometry and Groebner basis facts, plus the false injectivity assumption in Theorem 4.7 that breaks the advertised probability guarantees.

assumptions (4)
  • standard math Bertini's theorem and Rabinowitz trick: for general Theta, Lambda, mu, the ideal I_i(Theta,Lambda,mu) is unit or zero-dimensional, and g_i(X,C^n) equals dim_Q(R[T]/I_i).
    Used in Theorem 4.1; the proof states it follows from Bertini's Theorem and the Rabinowitz trick as in [24, Thm. 4.1].
  • standard math Pauer lucky prime framework: a prime avoiding all leading coefficients of a Groebner basis over Z yields the same leading monomials over Q and F_p.
    Basis of the modular computations; cited from [33, Prop. 4.1] and [4].
  • standard math Schwartz-Zippel lemma bounds the probability that a bounded-degree polynomial vanishes on random inputs.
    Used in Proposition 4.13 to convert degree bounds on the discriminant into probability bounds.
  • ad hoc to paper Nonconstant polynomials over F_p are injective, so a leading coefficient is nonzero with probability at least (p-1)/p.
    Assumed in the proof of Theorem 4.7; this is false, so the theorem's probability bound does not follow.

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Cite this review

Pith. "Pith review of Probabilistic Saturations and Alt's Problem." pith.science (2026). https://pith.science/paper/G2JT7M7W

@misc{pith2026190806020,
  author       = {Pith},
  title        = {Pith review of: Probabilistic Saturations and Alt's Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G2JT7M7W}},
  note         = {Machine review of arXiv:1908.06020}
}
read the original abstract

Alt's problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt's problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations. We give bounds on the size of finite field required to achieve a desired level of certainty. These methods can also be applied to many other problems where similar systems arise such as computing the volumes of Newton-Okounkov bodies and computing intersection theoretic invariants including Euler characteristics, Chern classes, and Segre classes.

Figures

Figures reproduced from arXiv: 1908.06020 by the authors.

Figure 1
Figure 1. A four-bar linkage with corresponding coupler curve 2.1. Relation to Alt’s problem. A four-bar linkage is the simplest moveable planar closed￾chain linkage, one of which is shown in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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