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I've got a sequence of polynomials, and for each of them the roots obviously follow a definite pattern. Here are the roots of the 34th one

enter image description here

All others have their roots arranged in a similar trident-like shape.

The Magma calculator says that for all of them the Galois group is the full symmetric group (well, I've checked some at random, including this one).

Are there any general methods to deal with such cases? From the picture I would guess that there is (at least) some cyclic group of order 3 involved somehow, so I should be able to reduce the polynomial to some simpler pieces.

In case anybody needs it, here is the 34th polynomial itself:

1262647690+255700718588*x+13631224452890*x^2+308600267562954*x^3+3762852287730239*x^4+28505564353529723*x^5+147540954575690309*x^6+558174350534761902*x^7+1622640970960835388*x^8+3765746767401417227*x^9+7187405631689627039*x^10+11550206297273077580*x^11+15923705986345125919*x^12+19119250849681784936*x^13+20236485734134957625*x^14+19066890500802679414*x^15+16118028554700214562*x^16+12301011609949474371*x^17+8516914887762136145*x^18+5369486377372114741*x^19+3090436418891829689*x^20+1626416851942385301*x^21+783117607008746782*x^22+344842286920653399*x^23+138658742267629274*x^24+50768083784135219*x^25+16852561930994233*x^26+5039986223113741*x^27+1345771479765129*x^28+316733973856726*x^29+64474270662688*x^30+11025181048833*x^31+1508607262720*x^32+150323855360*x^33+8589934592*x^34

$$ 8589934592 x^{34}+150323855360 x^{33}+1508607262720 x^{32}+11025181048833 x^{31}+64474270662688 x^{30}+316733973856726 x^{29}+1345771479765129 x^{28}+5039986223113741 x^{27}+16852561930994233 x^{26}+50768083784135219 x^{25}+138658742267629274 x^{24}+344842286920653399 x^{23}+783117607008746782 x^{22}+1626416851942385301 x^{21}+3090436418891829689 x^{20}+5369486377372114741 x^{19}+8516914887762136145 x^{18}+12301011609949474371 x^{17}+16118028554700214562 x^{16}+19066890500802679414 x^{15}+20236485734134957625 x^{14}+19119250849681784936 x^{13}+15923705986345125919 x^{12}+11550206297273077580 x^{11}+7187405631689627039 x^{10}+3765746767401417227 x^9+1622640970960835388 x^8+558174350534761902 x^7+147540954575690309 x^6+28505564353529723 x^5+3762852287730239 x^4+308600267562954 x^3+13631224452890 x^2+255700718588 x+1262647690 $$

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  • $\begingroup$ You say you expect to reduce the polynomial because of cyclic group of order 3. What exactly do you mean by reducing? Your polynomials are not irreducible? And your plot needs info on how to interpret: are the dots hights represent roots (real numbers always?) Multiple dots at same level does that means roots of same modulus? $\endgroup$ Apr 15, 2015 at 6:48
  • $\begingroup$ @PVanchinathan Sorry I should be less cryptic. These are the complex numbers, coordinates of the points are (real part, imaginary part). As for what exactly I mean by reducing - well unfortunately I am not sure what I mean. Maybe expressing them like $p(q(x))+r(x)$ where $p$, $q$ and $r$ are polynomials of lower degree. All I can claim is that there is some obvious third order symmetric pattern which I would like to make advantage of somehow. $\endgroup$ Apr 15, 2015 at 6:51
  • $\begingroup$ Can I assume all your polynomials are irreducible? $\endgroup$ Apr 15, 2015 at 6:55
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    $\begingroup$ I didn't see the picture 1st time I looked, it was not there (perhaps loading slowly). Now I see it - but all I see on it is what you expect from a polynomial with real coefficients - the symmetry w.r.t. the real axis. $\endgroup$ Apr 15, 2015 at 8:12
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    $\begingroup$ More precisely, I can say that the roots seem to lie on a kind of smooth (in each half-plane) curves. This certainly won't imply any symmetry in the usual sense. $\endgroup$ Apr 15, 2015 at 8:17

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