Auxiliary polynomial theorem

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

tatement

Let β equal the cube root of "b/a" in the equation "ax"³ + "bx"³ = "c" and assume "m" is and integer that satisfies "m" + 1 > 2"n"/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

: $max\_\{0\; le\; i\; le\; m+n\}\; \{(|u\_i|,|v\_i|)\}le\; 2b^\{9(m+n)\}$

References

* [http://ocw.mit.edu/OcwWeb/Mathematics/18-704Fall-2004/LectureNotes/index.htm lecture 31 MIT OCW]