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"Starting degree 5 it is proven that no analytical solution can exist and we must rely on trial and error methods." Not quite. No solution exists that uses a v

"Starting degree 5 it is proven that no analytical solution can exist and we must rely on trial and error methods." Not quite. No solution exists that uses a very restricted set of functions deemed "closed-form", in particular sums, products, quotients and n'th radicals. These polynomials can still be solved analytically and in closed-form by introducing some additional special functions. Certainly there are many ways forward rather than "trial and error methods," which I interpret as meaning the answer must be approximated numerically. For instance, all polynomials can be solved analytically and in closed form if elliptic functions are allowed to exist in the solution. Or as a simpler example, the quintic x⁵+x+a has no closed form solution in radicals, and is in some sense the prototypical example of an unsolvable quintic. This polynomial has one unique real root, which is a function of the variable a. In fact, we can look at this root as a function of a; this is called the "Bring radical" of a, denoted BR(a), and easily see it is a very simple, well-behaved, and perfectly "valid" analytic function of a. It is in some sense a sister function to the ordinary real fifth root of a, denoted ⁵√a, which would instead be the unique real root of the polynomial x⁵-a. The two are also just about equally easy to compute numerically to arbitrary precision. 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. This somewhat important detail is often omitted when explaining this. People often write the idea that there "is no general solution" to the quintic, which is misleading at best or wrong at worst. Perhaps a good analogy is to think back to the days when all numbers were thought to be rational. In such a system, even quadratic polynomials "have no analytic solution", one needs to invent the square root function to express the roots of the quadratic in closed form. The same thing happens for cubic and quartic polynomials, where we add cube roots and fourth roots; it just so happens for quintics we need to add two additional functions rather than one.

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.

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.