2

I am working on a program that uses a combination of heuristics and brute-force search to try to find solutions of polynomials of degree $>4$ in radicals. Although not all such polynomials are solvable, it turns out that many polynomials naturally occurring in the theory of special functions are actually solvable, and my program was able to find those solutions. See, for example, https://math.stackexchange.com/a/3345906/19661 and https://math.stackexchange.com/a/3353303/19661.

Could you please provide some other naturally occurring polynomials that you suspect might be solvable in radicals, so that I could test and improve my program. My intention is to open-source it once it is stable enough.

Vladimir Reshetnikov
  • 47,122
  • I can understand a vote to close, but I could not find a better way and audience to solicit this information. – Vladimir Reshetnikov Sep 24 '19 at 18:45
  • 1
    "Many polynomials naturally occurring in the theory of special functions are actually solvable" you are quite wrong, it is because you are looking at polynomials coming from special values of modular functions and modular forms with algebraic coefficients, all of those split completely in abelian extensions of $\Bbb{Q}(\sqrt{-d})$ (it is the theory of elliptic curves with complex multiplication) – reuns Sep 24 '19 at 18:47
  • @reuns Thanks. Yes, I understand that this is specific to certain classes of special functions. I am looking for examples where polynomials are suspected to be solvable in radicals, but no such solution has been found yet. – Vladimir Reshetnikov Sep 24 '19 at 20:16
  • 1
    For arbitrary solvable polynomials, the naive algorithm is to compute the normal closure (by factoring the polynomial repeatedly in extensions obtained by adding its roots) and its Galois group, finding a sequence of quotients making it solvable, thus obtaining a tower of cyclic extensions having radical closed-forms. When the polynomial comes from modular forms this procedure has a concrete meaning in term of the ideal class group of an order in $O_{\Bbb{Q}(\sqrt{-d})}$. – reuns Sep 24 '19 at 20:34

1 Answers1

2

I don't know about "naturally occurring", but you might try to generate some randomly.

For example, the following Maple code produced $3\,{x}^{6}-3\,{x}^{5}-4\,{x}^{4}-6\,{x}^{3}+3\,{x}^{2}+3$.

with(GroupTheory):
for i from 1 do
  p:= randpoly(x,degree=6,coeffs=rand(-10..10));
  if degree(p,x) < 6 then next fi;
  if not irreduc(p) then next fi;
  if IsSoluble(GaloisGroup(p,x)) then print(p); break fi
od:

EDIT: Another approach is to start with an expression in radicals, and find its minimal polynomial over the rationals. Maple can do this using evala(Norm(expression - x)).

q:= convert(sqrt(3^(1/2)+5^(1/3))+sqrt(-3^(1/2)+5^(1/3)),RootOf):
factor(evala(Norm(q-z)));

$$ \left( {z}^{12}+36\,{z}^{8}-320\,{z}^{6}+432\,{z}^{4}+1728 \right) ^{ 4} $$ So the minimal polynomial is $z^{12} + 36 z^8 - 320 z^6 + 432 z^4 + 1728$.

Robert Israel
  • 448,999
  • And solvable polynomials are as easy to produce as towers of radical extensions – reuns Sep 24 '19 at 18:58
  • 1
    The smallest real root is $x_1=\frac{1}{18} \left(3-\sqrt{57}+\sqrt[3]{12,\alpha}+\sqrt[3]{2/\alpha} \left(11\cdot \sqrt[3]9-3,\sqrt[6]{3} \cdot\sqrt{19}\right)\right),$ where $\alpha=288-7 \sqrt{57}+9 \sqrt{999-42 \sqrt{57}}.$ I believe there is a known systematic way to solve all solvable polynomials up to degree $7$. – Vladimir Reshetnikov Sep 24 '19 at 19:06
  • 1
    Quintics, sextics, septics. BTW, do you know if any of these algorithms are implemented in Maple? – Vladimir Reshetnikov Sep 24 '19 at 19:25
  • In general Maple's solve command doesn't seem to go higher than quartics (or polynomials that are easily reducible to quartics). – Robert Israel Sep 24 '19 at 21:21
  • 1
    For an octic try ${x}^{8}+4,{x}^{7}+8,{x}^{6}+7,{x}^{4}-4,x-7$. – Robert Israel Sep 24 '19 at 21:25
  • Thanks. The octic factors into quartics in $\mathbb Q(\sqrt2)$. My program tries simple cases like this first. – Vladimir Reshetnikov Sep 24 '19 at 21:33
  • Unfortunately Maple gives Error, (in GroupTheory:-GaloisGroup) cannot handle polynomials of degree higher than 9 for higher degrees (I just installed a trial version). – Vladimir Reshetnikov Sep 24 '19 at 21:45
  • Yes, GaloisGroup (and the earlier command galois) are limited to degree $\le 9$ for univariate polynomials. You might try Magma (I have not, but supposedly it can compute Galois groups for polynomials of arbitrary degree). – Robert Israel Sep 25 '19 at 01:12