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The problem can be solved by running some Integer Relation algorithm (e.g., PSLQ) on the numbers $1, r, r^2, \dots, r^N$ where $r$ is a given root.

For example, here is computation in PARI/GP which gives a better result than the polynomial shown in question:

? r = 28.552622898861801; algdep(r,10)

%1 = 3*x^10 + 38*x^9 - 3695*x^8 + 4582*x^7 + 3016*x^6 + 1435*x^5 + 4552*x^4 - 1219*x^3 - 9920*x^2 - 2402*x + 3087

? subst(%1,x,r)

%2 = -2.7334689816478450022 E-24

1

The problem can be solved by running some Integer Relation algorithm (e.g., PSLQ) on the numbers $1, r, r^2, \dots, r^N$ where $r$ is a given root.