Wildberger's Polynomial Breakthrough Meets Zeilberger's Algorithmic Proofs

Norman Wildberger's May 2025 paper introducing a hyper-Catalan series solution to polynomial equations has sparked a direct and fruitful collaboration with the Wilf-Zeilberger algorithmic proving community. Within weeks of publication, Doron Zeilberger and collaborators applied WZ techniques to prove Wildberger's conjectures about the newly discovered "Geode" array, demonstrating a compelling intersection of constructivist algebra and automated theorem proving.

The Wildberger-Rubine polynomial result

Publication details: "A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode" by Norman J. Wildberger and Dean Rubine appeared in the American Mathematical Monthly, Volume 132, Number 5, pages 383-402 (published online April 8, 2025; print May 2025). The paper is open access under Creative Commons CC BY, with DOI: 10.1080/00029890.2025.2460966.

The core innovation extends classical Catalan numbers—which count ways to triangulate polygons and relate to quadratic equations—into hyper-Catalan numbers C[m₂, m₃, m₄, ...] that count subdivisions of "roofed polygons" into triangles, quadrilaterals, pentagons, and so on. The precise factorial formula is:

C[m₂, m₃, m₄, ...] = (c₁ + m₂ + 2m₃ + 3m₄ + ...)! / [(c₁ + 1)! · m₂! · m₃! · m₄! · ...]

where c₁ = 1 + m₂ + m₃ + m₄ + ... represents the number of internal faces plus one.

For any polynomial equation c₀ + c₁x + c₂x² + c₃x³ + ... = 0, Wildberger and Rubine provide a power series solution:

x = Σ C[m₂, m₃, m₄, ...] · c₀^(1+m₂+2m₃+...) · (c₂/c₁²)^m₂ · (c₃/c₁³)^m₃ · ...

This does not contradict the Abel-Ruffini theorem or Galois theory, which prove no finite radical formula exists for polynomials of degree five or higher. Instead, Wildberger's approach sidesteps radicals entirely using infinite power series that can be truncated for arbitrarily precise approximations. The method recovers Eisenstein's 1844 series for the Bring radical quintic as a special case and connects to Lagrange's series reversion formula.

Zeilberger applies WZ methods to prove the Geode conjectures

The most significant development is Zeilberger's direct engagement. Within two months of publication, Tewodros Amdeberhan and Doron Zeilberger submitted "Proofs of Three Geode Conjectures" (arXiv:2506.17862, June 22, 2025), published in Enumerative Combinatorics and Applications, Volume 6:1 (2026), Article #S2R2.

The Geode is a mysterious multi-dimensional array G that Wildberger discovered arising from the factorization S - 1 = S₁ · G, where S is the hyper-Catalan generating series. Wildberger posed three conjectures about this array, all of which Amdeberhan and Zeilberger proved using:

• Wilf-Zeilberger algorithmic proof theory as the primary tool
• Constant-term extraction
• Lagrange inversion
• The multinomial theorem

The paper explicitly constructs WZ certificates for key identities. For example, for the 2D Geode closed form (Theorem 1.1), they employed the WZ pair:

• F(n,k) = (-1)^k · C(n,k) · C(2n+1+k, n+1+k) / (2n+1+k)
• H(n,k) = -F(n,k) · k(n+1+k) / (n(2n+1))

These satisfy the fundamental WZ relation F(n,k) = H(n,k+1) - H(n,k), enabling automatic proof of the identity.

Closed-form results obtained through WZ methods

The WZ approach yielded explicit closed-form expressions for special cases of Geode entries:

Theorem 1.1 (Bi-Tri case): G[m₁,m₂] = (2m₁+3m₂+3)! / [(2m₁+2m₂+3)(m₁+m₂+1)(m₁+2m₂+2)!m₁!m₂!]

Theorem 1.2 (Consecutive types): A generalized formula for G[0,...,0,mₐ,mₐ₊₁] with parameter a

Theorem 1.3: A formula for alternating indices G[-f,f,...,-f,f]

The Amdeberhan-Zeilberger paper describes Wildberger's work as "fascinating" and uses it as a launching point for further exploration using WZ machinery.

Computational challenges and the holonomic ansatz

A second Zeilberger paper, "The Challenge of Computing Geode Numbers" with Amdeberhan and Manuel Kauers (arXiv:2508.10245, August 2025, to appear in Palestine Journal of Mathematics), applied the holonomic ansatz to Geode computation. They discovered that while the 2D case has nice closed forms, the 3D case required combining experimental mathematics with holonomic methods, and the 4D case posed severe computational difficulties.

The authors offered $100-$200 OEIS donations as prizes for computing specific Geode values like G[1000,1000,1000,1000]. Dean Rubine claimed this prize in December 2025 (arXiv:2512.21785), using hyper-Catalan recurrences with optimized caching to obtain a 6,303-digit result for the 4D Geode diagonal.

Combinatorial interpretations from other researchers

The mathematical community responded vigorously with follow-up work providing deeper understanding:

• Ira M. Gessel (Brandeis University, arXiv:2507.09405) proved that Geode coefficients count nonnegative lattice paths, connecting the Geode to standard enumerative combinatorics
• Fern Gossow (arXiv:2507.18097) provided an interpretation in terms of ordered trees, though importantly disproved one of Wildberger's original conjectures about what Geode numbers count while providing a corrected interpretation
• Dean Rubine (arXiv:2507.04552) independently derived recurrences for hyper-Catalan numbers and proved Wildberger's conjectures through elementary methods

The philosophical connection between Wildberger and Zeilberger

Both mathematicians share finitist or ultra-finitist sympathies. Wildberger famously rejects irrational numbers, arguing that expressions like ∛7 cannot be "fully calculated" as complete objects. Zeilberger has stated: "I prefer finite mathematics much more than infinite mathematics... Infinity mathematics, to me, is something that is meaningless."

However, Zeilberger's engagement with Wildberger's work has been purely mathematical rather than philosophical, focusing on proving conjectures and developing computational methods without discussing foundational issues. No dedicated "Opinion" about Wildberger appears on Zeilberger's famous opinions page (which contained 193 numbered opinions as of December 2025).

Academic reception and media coverage

The result received extensive popular science coverage from Phys.org, Newsweek, Popular Science, Live Science, ScienceAlert, and others, though professional mathematicians noted that headlines like "Mathematician Solves Algebra's Oldest Problem" significantly overstated the novelty. The method represents an elegant synthesis of Lagrange inversion with combinatorial structures rather than a revolutionary overthrow of Galois theory.

The consensus among mathematicians discussing the work is that it constitutes a legitimate contribution to combinatorics and algebra, particularly through the discovery of the Geode array, which has generated substantial research interest. The American Mathematical Monthly—a respected MAA journal—published the peer-reviewed result, and the follow-up work by established combinatorialists like Zeilberger and Gessel validates its significance.

Conclusion

The intersection of Wildberger's polynomial-solving framework with Wilf-Zeilberger automated proving represents an active and productive research area. The key finding is that WZ methods have been directly and successfully applied to prove Wildberger's Geode conjectures, with explicit WZ certificates constructed for factorial identities arising from hyper-Catalan numbers. The holonomic ansatz has facilitated Geode computation, and closed-form expressions have been obtained for special cases. A general closed form for arbitrary Geode entries remains an open problem, representing a natural target for future applications of creative telescoping and hypergeometric identity methods. The rapid 2025-2026 activity demonstrates how traditional power series methods, when reframed through combinatorial structures, can generate entirely new computational and proof-theoretic challenges that automated methods are well-suited to address.