3 ms·

> If we allow BR(a) to "exist" as a function worthy of being deemed "closed-form," "analytic" or what have you, then every quintic now has a closed form solutio

by feoren 4y ago

> If we allow BR(a) to "exist" as a function worthy of being deemed "closed-form," "analytic" or what have you, then every quintic now has a closed form solution in terms of radicals and this one extra special function. At first I thought this was smoke and mirrors: define a function R(a, b, c, d, e, f) to be the (say, smallest) root of ax^5 + bx^4 + cx^3 + dx^2 + ex + f and now have a solution to every quintic (if we allow R in your solution)! But your point about BR(a) being about as easy to numerically compute as ⁵√a is really where you started to change my mind, because of course that's what we end up caring about if we want to solve any real-world problem that involves finding the roots of a quintic. Are you saying that we could write down one unique "quintic formula" that is the same for every quintic, using sums, products, quotients, n'th radicals, and BR? If so, this is a really important point that everyone who studies polynomials should understand. Or do you still need some "trial and error" to reduce an arbitrary quintic to a form that can be solved using BR?

ComplexSystems 4y ago
Yes, using a purely mechanical process every quintic can be solved in closed form this way. The usual method isn't all that different from the way the quartic is solved. You do a change of variables to eliminate the x⁴ term, and then a few more simple changes of variables will get it to the Bring normal form, for which the solution is the Bring Radical. Then you can unwind the changes of variables to get a closed form expression in the original variable x. Each step is totally mechanical and involves no "trial and error" at all. The only issue is that the resulting expression can be extremely large once you unwind the change of variables. It's kind of similar to taking (for instance) a small C++ program and explicitly expanding every macro or function with a copy of the entire function body, producing a huge space increase. You are usually better off leaving things in terms of the transformed variable, along with the formulas to convert back to the each previous variable. This is also an issue with the solution to the general quartic, though, so it's neither here nor there regarding Bring radicals.

by feoren 4y ago

ComplexSystems 4y ago

Yes, using a purely mechanical process every quintic can be solved in closed form this way. The usual method isn't all that different from the way the quartic is solved. You do a change of variables to eliminate the x⁴ term, and then a few more simple changes of variables will get it to the Bring normal form, for which the solution is the Bring Radical. Then you can unwind the changes of variables to get a closed form expression in the original variable x. Each step is totally mechanical and involves no "trial and error" at all. The only issue is that the resulting expression can be extremely large once you unwind the change of variables. It's kind of similar to taking (for instance) a small C++ program and explicitly expanding every macro or function with a copy of the entire function body, producing a huge space increase. You are usually better off leaving things in terms of the transformed variable, along with the formulas to convert back to the each previous variable. This is also an issue with the solution to the general quartic, though, so it's neither here nor there regarding Bring radicals.