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Abstract

ÌýThis is an exposition of work which recently appeared in the American Mathematical Monthly (May 2025), joint work with computer scientist Dean Rubine.Ìý

We show how to solve a general polynomial equation without radicals and Galois theory, relying rather on an earlier tradition involving series, developed by Newton, Leibniz, Euler and Lagrange.Ìý

To do this we extend the Catalan story from a sequence to a multidimensional array, initiate an algebra of subdivided planar roofed polygons (subdigons), and then connect with traditional algebra through an appropriate accounting function from multisets of subdigons to polynomials.

Working out the details connects us with a rich vein of combinatorics, along with Lagrange's reversion/inversion of series, and also somewhat surprisingly Euler's polytope formula.Ìý

The resulting multivariate generating function S (the solution!) has a remarkable factorization which reveals a previously unknown layer underneath Catalan numerics, which we call the "Geode". Many new questions and possible developments for research arise.Ìý

But we can also use this S directly to solve real life polynomial equations, as demonstrated on Wallis' famous cubic example. And we show finally how to really solve a quintic equation!

This talk will be (hopefully) understandable to first year students.

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