Auxiliary polynomial

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Auxiliary polynomials are an important construction in transcendental number theory.

[edit] Definition

For each 1 ≤ m ≤ n let Llm (1 ≤ l ≤ k) be the linear forms in x1m, …, xkm with real algebraic rational coefficients. Further let d be the degree of the field K generated by all the coefficients over the rationals. We shall signify by c1, c2, … the numbers greater than one which depend on these coefficients only. Now let r1, …, rn be any positive integers, and let r = max(rm). Further, suppose that 0 < є < 1 and that e(1/4)є2n > 2kd.

With these conditions in place:

There is a polynomial P with degree at most rm in x1m, ..., xkm and with height at most c_1^r such that for each l with 1 ≤ l ≤ k, the index of P with respect to the Llm and rm is at least n/k − єn.

It can be assumed that for all l, m, the coefficient of x1m in Llm, say αlm is not zero. Then P has to be determined such that, for all l and non-negative integers j1 + ... + jn with

\sum_{m=1}^n j_m/r_m < n/k-\varepsilon n,

the polynomials

(j_1!\cdots j_n!)^{-1}(\delta/\delta x_{11})^{j_1}\cdots(\delta/\delta x_{1n})^{j_n} P

vanish identically when −Llm, with x1m equated to zero, is substituted for x1m and the factor αlm is included to multiply each of x2m, ..., xkm. Now these polynomials are homogeneous in x2m, ..., xkm with degree rm − jm and hence they have in total, at most kNe(1/4)є2n coefficients, where N denotes the product of the binomial factors.

[edit] References

  • Baker, Alan; Transcendental Number Theory, 1975, Cambridge University Press, Great Britain, ISBN:0-521-20461-5