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Everyone knows that:

  1. quadratic parabola always has mirror geometric symmetry
  2. cubic parabola always has rotation geometric symmetry
  3. quartic parabola does not have in general (universal) geometric symmetry

Mathematicians know that there is general (universal) algebraic formula for the roots of all 2,3,and 4 power polynomials. This is based on symmetry of Galois groups of field extensions something related to root radicals (please correct me if I am wrong somewhere)

I am very curious if this geometric symmetry of graphs could be somehow represented in terms of Galois groups symmetry? Or these two symmetries are nor related?

Please, explain me, and if it is possible with some examples: like why 4 parabola is not generally symmetric geometrical but roots are symmetric algebraicaly etc. (of course if such explanation is possible and have sense)

Thank you!

Artem
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  • The symmetries of order three may manifest themselves as rotations of the complex plane (and/or involve the zeros of auxiliary polynomials). Of course, those cubics that can be solved using trig formulas also bring in that symmetry as a 120 degree rotation (possibly better thought of as a 120 degree phase shift). – Jyrki Lahtonen Apr 06 '22 at 04:27
  • Thank you, as I understand now the short answer is “no connections”. I have also found similar question with your comments and that was very useful and unformative: https://math.stackexchange.com/questions/2960586/symmetries-of-polynomials – Artem Apr 06 '22 at 04:52
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    It depends on what we count as a symmetry. What I meant with the trig formulas is shown (for example) in the following. Consider the cubic equation $$4x^3-3x=a,$$ where $a$ is a real constant with absolute value $<1$. The trick is that the substitution $x=\cos t$ turns the left hand side into $\cos 3t$, implying that the solutions come from $3t=\arccos a+2\pi n$, $n$ an integer (it suffices to consider $n=0,1,2$ only), and $x_n=\cos(\dfrac13[\arccos a+2\pi n])$. The phase shift by $2\pi/3$ permutes the roots cyclically $x_0\mapsto x_1\mapsto x_2\mapsto x_0$. – Jyrki Lahtonen Apr 06 '22 at 05:11
  • (cont'd) This won't show as a three-fold geometric symmetry in the same way that the reflectional symmetry of a parabola, swapping the roots. – Jyrki Lahtonen Apr 06 '22 at 05:14
  • Thank you for your time, so as I understood: these geometric parabolas features and root rotations symmetries are some sort of coincidences. Both could be called symmetries, both could be studied as a group theory object, both are related to polinomiala but nature of these two phenomena are different (first rotations/reflections of figure and second is permutations of roots) – Artem Apr 06 '22 at 05:33
  • Following your trigonometric substitution example and complex plane rotation symmetry I got a thought that some geometric and algebraic symmetry connection could be for complex parabolas (I checked for example cubic parabala and that is on a large range x -> infinity there are distinct symmetric regions) maybe this sort of connection could be formalised somehow and on the second step complex parabola graph could be linked with real one, but that is only my intuition and fantasies) – Artem Apr 06 '22 at 07:01

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