Mathematician solves algebra's oldest problem using intriguing number sequences

The actual article (OA!): <a href="https:&#x2F;&#x2F;www.tandfonline.com&#x2F;doi&#x2F;full&#x2F;10.1080&#x2F;00029890.2025.2460966" rel="nofollow">https:&#x2F;&#x2F;www.tandfonline.com&#x2F;doi&#x2F;full&#x2F;10.1080&#x2F;00029890.2025.2...</a><p>This is about solving polynomial equations using Lagrange inversion. This method, as one might have guessed, is due to... Lagrange. See <a href="https:&#x2F;&#x2F;www.numdam.org&#x2F;item&#x2F;RHM_1998__4_1_73_0.pdf" rel="nofollow">https:&#x2F;&#x2F;www.numdam.org&#x2F;item&#x2F;RHM_1998__4_1_73_0.pdf</a> for a historical survey. What Wildberger is suggesting is a new(?) formula for the coefficients of the resulting power series. Whether it is new I am not sure about -- Wildberger has been working in isolation from others in the field, which is already full of rediscoveries. Note that the method does not compete with solutions in radicals (as in the quadratic formula, Tartaglia, Cardano, del Ferro, Galois) because it produces infinite sums even when applied to quadratic equations.<p>Phys.org has gotten no part of the story correct.

Is this something with potential to change mathematics as taught in schoolbooks, or just some sort of galactic algorithm[1] or research niche that will never spread beyond the expertise domain of half a dozen research groups across the whole world?<p>[1] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Galactic_algorithm" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Galactic_algorithm</a>

So the mathematicians have found an algorithm for producing the roots of quintic and higher order polynomials in terms of power series? Normally one would find these roots using an iterative algorithm so this seems like a big deal.

I don&#x27;t get this. If they are using power series, aren&#x27;t they in practice still getting an approximation of the result anyway? Why would this be a better way of approximating the solution of a polynomial equation?

&gt; So, when we assume ∛7 &quot;exists&quot; in a formula, we&#x27;re assuming that this infinite, never-ending decimal is somehow a complete object.<p>&gt; This is why, Prof. Wildberger says he &quot;doesn&#x27;t believe in irrational numbers.&quot;<p>OK. I’m with you.<p>&gt; His new method to solve polynomials also avoids radicals and irrational numbers, relying instead on special extensions of polynomials called &quot;power series,&quot; which can have an infinite number of terms with the powers of x.<p>So, he believes this other infinite, never ending thing exists? Isn’t that a tad inconsistent?

I think the following paper from 2016 contains the same idea already: <a href="https:&#x2F;&#x2F;arxiv.org&#x2F;abs&#x2F;1607.04144" rel="nofollow">https:&#x2F;&#x2F;arxiv.org&#x2F;abs&#x2F;1607.04144</a>

Non-AMP link: <a href="https:&#x2F;&#x2F;phys.org&#x2F;news&#x2F;2025-05-mathematician-algebra-oldest-problem-intriguing.html" rel="nofollow">https:&#x2F;&#x2F;phys.org&#x2F;news&#x2F;2025-05-mathematician-algebra-oldest-p...</a>

The Greeks fought √2 to the last Pythagoreans standing but √2 won, so best of luck to prof Wildberger.

Do I understand correctly that this gives a formula to solve polynomic equations, which needs no square roots, but it also doesn‘t need complex numbers?

&gt; This is why, Prof. Wildberger says he &quot;doesn&#x27;t believe in irrational numbers.&quot;<p>Oh boy, I hope that they missed a joke or misquoted.

I&#x27;m a mathematics professor, specializing in number theory.<p>As others have noted (the author apparently &quot;doesn&#x27;t believe in irrational numbers&quot;), this press release is laughable and utterly absurd. Wildberger did not &quot;solve algebra&#x27;s oldest problem&quot;, or anything remotely close to that.<p>I checked out phys.org -- I assumed this would be the webpage of some prominent national society or something -- but it turns out to be some randos that have a publishing outfit.<p>I did, however, look up the original paper. Unfortunately it seems to be paywalled, although I have access through my university.<p>The actual paper seems to for the most part be sober, legitimate, and potentially interesting (albeit on the same scale that many many other published math papers are interesting). Except for a bit of hyperbole in the introduction, it doesn&#x27;t traffic in exaggerated claims. Seems to be a legitimate effort, somewhat off the beaten track.