I have an expression for 2 algebraic numbers $b,c$ in a complicated polynomial:
$S(b,c) = 2.863970296b^{16} - 34.31251041b^{15} + ( -8.619857140c^2 + 164.3692754)b^{14} + (36.88920253c^2 - 340.4736152)b^{13} + ( -16.36994915c^4 + 162.2208545c^2 - 11.45480546)b^{12} + (190.1812378c^4 - 1132.476736c^2 + 1464.184063)b^{11} + (25.88724497c^6 - 421.6383919c^4 + 1717.603903c^2 - 2578.258770)b^{10} + (68.85869275c^6 - 744.6009089c^4 + 1538.701443c^2 + 598.2060072)b^9 + (58.40384598c^8 - 684.2811091c^6 + 3422.863457c^4 - 6926.863835c^2 + 3256.986225)b^8 + ( -113.7069658c^8 + 574.5101623c^6 - 2053.658235c^4 + 5454.600718c^2 - 3834.980888)b^7 + (33.94926789c^{10} - 316.2058102c^8 + 2002.061976c^6 - 4872.708215c^4 + 2994.879534c^2 + 467.9336210)b^6 + ( -63.58613369c^{10} + 707.9851930c^8 - 3192.568248c^6 + 6870.315358c^4 - 6411.657219c^2 + 1928.106174)b^5 + (6.318788606c^{12} - 67.64939688c^{10} + 249.1303828c^8 - 255.6473521c^6 - 783.7149856c^4 + 2191.500177c^2 - 1349.816178)b^4 + (186.5453734c^{10} - 1129.575896c^8 + 2478.081049c^6 - 2551.792526c^4 + 833.5822206c^2 + 218.0152376)b^3 + ( -16.23472475c^{12} - 42.71980675c^{10} + 378.4386754c^8 - 562.0173472c^6 + 428.2871476c^4 - 249.0050575c^2 + 54.41258233)b^2 + (8.992868848c^{12} - 76.15597847c^{10} + 316.7598470c^8 - 763.8661377c^6 + 833.4686952c^4 - 319.6396293c^2 + 1.255532031)b + (2.721641067c^{12} + 29.02631922c^{10} - 151.3837260c^8 + 309.0143681c^6 - 300.6269619c^4 + 118.2842809c^2 - 7.035921417) = 0$
where the optimal values are close to $b = -0.3552741670464769056638520...$ and $c = \pm 0.6153529079412105375715555...$ as solutions to $S(b,c)=0$. I have previously determined the decimal value of $b$ to 5009 digits and $c$ to 20,036 digits and they are algebraic.
The numeric coefficients to $S(b,c)$ above are from polynomials obtained from a certain root of a 104th degree equation, which minimal polynomial I have. That root is close to $0.7105483340929538113277041...$ I did not choose to show the full polynomial in 3 variables as that would have been almost unreadable in this forum.
Question:
Can the minimal polynomials for $b,c$ be recovered from $S(b,c)$? I suspect that the degree might be 832 or 1664 or higher (a multiple of 104).
Last night I was able to recover the minimal polynomial for b, and the polynomial above simplified to:
$-2.4217682756002569515269958118100226692c^{12} + 41.675721951368927271613130255284325166c^{10} - 166.07875566229344222912103497760334885c^8 + 415.76852679745199779032163873461583049c^6 - 487.03238625701722288392965721478730476c^4 + 234.53560111758265534390801715118928961c^2 - 38.453353800469690586623289970241113230 = 0$
where the numeric coefficients are from the algebraic numbers $a,b$.
Newer question: How can I recover the minimal polynomial for $c$?
The key term you need is integer relation finding algorithm. The classic one is LLL, and the most popular is probably PSLQ. They will find a relation between the first n powers of a number which is known to high precision, which is effectively finding the minimum polynomial.