Hyper-Catalan

This might be the biggest news in arithmetic computation in my lifetime: N. J. Wildberger and Dean Rubine, "A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode". It's all over the place, like ScienceDaily.

Polynomial equations are ax^2+bx+c=y; the notion is, y=0 solve for x. This here was a "second order" equation, since up to some x²; n=2. For such, exist up to n solutions. For the second-orders, called "quadratic": Sargon of Akkad might not have known, but Hammurabi sure did. That's how old it is. The solution of x^2-2=0 could be shorthanded as its "radical", root-two here; the Greeks found that was as good as they could do, since it couldn't be expressed as a fraction (or a decimal, which is just x/10^n you know). The Greeks then found that no equation could encompass π; it wasn't algebraic.

It took until the gunpowder-and-caravel age, but then some Italians came up with good equations to dig out x in third- and fourth-order polynomials, called "cubics" and "quartics". Those solutions could be expressed as radicals like ol' root-2. There was actually something of a rivalry in Italy between cities as could solve these secret equations and those as hoped to. Nobody, however, could suss out quintics beyond "up to five solutions, real or complex", barring blatantly artificial concoctions like x^5-2=0. The Newton-Raphson method and maybe some others can approximate solutions but they're algorithmic. (Remember that π was already off this table, these mathematicians didn't bother with that.)

Abel, Ruffini, and Galois in the late date of 1822 finally proved that radicals cannot always be found to express an algebraic number. However in 1844, Gotthold Eisenstein floated x^5+x-t=0; although the x quintet of solutions ain't radical, such can at least be expressed - as a powerseries. If the decimal numbers trail off into infinity who cares; same holds for the two-radical (squareroot) of two. Just like for the "Genus One" ellipse which, incidentally, isn't even polynomial.

That is where Wildberger and Rubine step in, two centuries later. They are expanding Eisenstein's nonradical expressions to cover the whole much of the plane of polynomial solutions. No more Newton-Raphson: want the solution? just go crunch. In retrospect, we should have been able to do this before we did the ellipse, but then... the low-genus curve was being used for crypto.

INTERJECT 5/21 As provisos go, of course we are anchoring the coëfficients, in c + c₁x + ..., such that c not zero. I mean, duh: if c zero in a quintic then what you got is not a quintic, but a quartic (where x isn't zero itself). Less-intuitively: if c != 0 also c₁ can't be zero.

That could be big for repetitive calculations in 3D graphics and astronomy. We'll see. SEEN 5/21: Your laptop can't do it. Sorry to repeat the bad news in this, about the most-viewed post in recent history of this blog.