# Let's check where 17/324 and 5/54 come from: # Could they ...

Prompt

# Let's check where 17/324 and 5/54 come from: # Could they come from inverting a map, or from an approximate root with respect to Y? # Let's consider P and Q as polynomials in Y with coefficients in C[X]: # P = (X^51) Y^51 - (1/2 X^41) Y^13 + X # Q = (X^85) Y^85 + (5/6 X^75) Y^47 - (5/18 X^65) Y^9 - (5/54 X^55) Y^9 + (17/324 X^15) Y^22 + Y # # Wait! In Q, look at the coefficients: # coeff of Y^9 is: -5/18 X^65 - 5/54 X^55 = -5/18 X^55 (X^10 + 1/3) # WAIT! LOOK AT THAT: # -5/18 X^65 - 5/54 X^55 = -5/54 X^55 (3 X^10 + 1) !! # = -5/18 X^65 - 5/54 X^55 = -5/54 X^55 * (3 X^10 + 1) # = -5/18 X^65 - 5/54 X^55 = -5/18 X^65 (1 + 1/(3 X^10)) # WOW! print(Rational(-5, 18), Rational(-5, 54)) print(Rational(-5, 18) / Rational(-5, 54))