Auxiliary polynomial theorem

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The construction of auxiliary polynomials is an important concept in diophantine approximation and transcendental number theory.

[edit] Statement

Let β equal the cube root of b/a in the equation ax3 + bx3 = c and assume m is and integer that satisfies m + 1 > 2n/3 ≥ m ≥ 3 where n is a positive integer.

Then there exists

F(X,Y) = P(X) + Y * Q(X)

such that

\sum_{i=0}^{m+n} u_i X^i = P(X),
\sum_{i=0}^{m+n} v_i X^i = Q(X).

The auxiliary polynomial theorem states

\max_{0 \le i \le m+n} {(|u_i|,|v_i|)}\le 2b^{9(m+n)}

[edit] References