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An explicit version of Birch’s Theorem. (English) Zbl 0915.11021

For \(1 \leq i \leq r\) let \(f_i({\mathbf x})\) be a system of homogeneous polynomials of degree \(d\) over the rationals in \(s\) variables \({\mathbf x}=(x_1,\dots x_s)\). If \(s\) exceeds a certain bound \(\nu=\nu_{\smash {d,r}}^{(m)}\) one wishes to assert that the system of equations \(f_i({\mathbf x})=0\) possesses a solution set of projective dimension \(m\). B. J. Birch’s theorem [Mathematika 4, 102-105 (1957; Zbl 0081.04501)] was to the effect that when \(d\) is odd such a \(\nu\) exists. The author obtains an explicit upper bound for the least such \(\nu\).
For a given increasing function \(\Psi\) let \(\Psi_x\) denote its \([x]\)-fold iterate. Then let \[ \psi^{(n)}(x)=\smash{\psi_{42\log x}^{(n-1)}(x)} \] with \(\psi^{(0)}(x)=e^x\). Then the author’s rather large bound is \(\psi^{(d-5)/2}\bigl( dr(m+1) \bigr)\), provided \(d>5\).
The iterative methods used depend on an “efficient diagonalisation” process used by the author in [T. D. Wooley, Analytic number theory, Lond. Math. Soc. Lect. Note Ser. 247, 361-376 (1997; Zbl 0905.11022)] in a discussion of the case of forms of degree \(d=3\) or \(5\).
The author remarks that Birch’s method could, in principle, be developed to yield an explicit bound in this problem that would, however, be extraordinarily weaker than the result in this paper.
Another approach that could yield an explicit bound is the version of the circle method developed by W. M. Schmidt [Acta Math. 154, 243-296 (1985; Zbl 0561.10010)]. The author provides a discussion which indicates that Schmidt’s method would lead to weaker bounds than his own as soon as \(d\geq 7\) and \(r\) is large.

MSC:

11D72 Diophantine equations in many variables
11E76 Forms of degree higher than two
11G35 Varieties over global fields
14G05 Rational points
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