I have been developing a procedure for computing the polynomial associated with a given eccentric planetary gear train. In outline, it generates sunsets by solving inverse trigonometric equations of form belt-length = integer , numerically to high precision; then deduces an integer relation between successive powers of some chosen sunset, via LLL lattice basis reduction. The process is not currently fully automated: the user remains obliged to estimate suitable precision and polynomial degree, and to select a sunset (often but not always the minimum) not subsumed by some smaller train. As an example, consider the (symmetric) train [r, s; p, t, q] = [28, -12; 12, 8, 4] = 4*[7, -3; 3, 2, 1] . Computing possible sunsets h , then processing h -> h^2 * 2/(r-s) = h^2/20 (to reduce eventual polynomial degree and coefficient size) yields values to 6 decimal digits [0.164185, 0.177674, 0.203751, 0.250000, 0.334135, 0.504112, 0.937348, 4.00000] . (Maximum value 4 represents sun and ring touching externally: an overlapping, numerically pathological case best excluded!) Because this train is (concentric with a multiple of) 4x a smaller train, alternate values merely yield polynomials associated with 2x or 1x multiples. But applying LLL to powers of (say) the first value yields polynomial 400*X^6 + 768*X^5 + 22425*X^4 - 38960*X^3 + 19680*X^2 - 3840*X + 256 , with just 4 positive real roots, corresponding to the 4 remaining values. Its coefficients are highly composite: [2^4 5^2, 2^8 3, 3 5^2 13 23, -2^4 5 487, 2^5 3 5 41, -2^8 3 5, 2^8] . Substituting all 8 values into the polynomial yields syndrome [2.5 E-179, -1.0, -3.6 E-179, 5.7, -1.3 E-179, -1.9 E2, 2.5 E-177, 6.0 E6] . Working single precision was 180 digits, double that for solving equations and accumulating inner products; degree assumed was m = 12 ; polynomial factors of degrees 1,5 were dropped from the raw result. A remarkable feature: the smaller multiple [21, -9; 9, 6, 3] = 3*[7, -3; 3, 2, 1] yields processed values h^2/15 = [0.167544, 0.193318, 0.250000, 0.377104, 0.736875, 4.00000] (4 relevant), and polynomial 400*X^6 + 3684*X^5 + 22425*X^4 - 38960*X^3 + 19680*X^2 - 3840*X + 256 --- identical to that above except for a single coefficient! QUERY: Given dimension (ie. presumed polynomial degree) m , what precision is necessary to ensure successful lattice basis reduction? Fred Lunnon