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**A proof of Pisot’s \(d\)th root conjecture.**
*(English)*
Zbl 0995.11007

It is straightforward that if \(\sum_{h\geq 0} b_hX^h\) and \(\sum_{h\geq 0} c_hX^h\) are power series representing rational functions then their Hadamard product \(\sum_{h\geq 0} b_hc_hX^h\), and the Hadamard powers \(\sum_{h\geq 0} b_h^kX^h\), \(k=2\), \(3\), \(\ldots \), all are rational functions.

Conversely, suppose that \((a_h)_{h\geq 0}\) and \((b_h)_{h\geq 0}\) are sequences of rational integers so that both \(\sum a_hX^h\) and \(\sum b_hX^h\) represent rational functions. Pisot conjectured that if \(b_h \bigm |a_h\) for \(h=0\), \(1\), \(\ldots \) also the Hadamard quotient \(\sum (a_h/b_h)X^h\) represents a rational function; respectively, if, say, \(a_h=c_h^3\), \(h=0\), \(1\), \(\ldots \) for some sequence \(c_h\) of rational integers then also the Hadamard cube root \(\sum c_hX^h\) is rational. This paper proves the natural generalisation of Pisot’s root conjecture.

In proving the quotient result [see ‘Solution de la conjecture de Pisot sur le quotient de Hadamard de deux fractions rationnelles’, A. J. van der Poorten, C. R. Acad. Sci., Paris, Sér. I 306, 97-102 (1988; Zbl 0635.10007)] and subsequent related work, the present reviewer noted inter alia that the integrality conditions on the sequences may be replaced by the requirement that the sequences belong to a ring finitely generated over \(\mathbb Z\) and that one readily reduces that case to that of \(S\)-integers in some algebraic number field. In that context the Hadamard \(k\)-th root conjecture asks only that the \(a_h\) all be \(k\)-th powers in some field \(\mathbf F\) finitely generated over \(\mathbb Q\). Indeed, given that, in the present paper it is proved that there then is a sequence \((c_h)\) with \(c^k_h=a_h\) for \(h=0\), \(1\), \(\ldots \) so that \(\sum c_hX^h\) is rational. Strikingly, the author’s methods involve ideas noticeably different from those applied in the proof of the quotient theorem.

Of course it is well known that if \(\sum a_hX^h\) is rational then the coefficients \(a_h=a(h)\) are given by a generalised power sum \(\sum_{i=1}^m A_i(h)\alpha_i^h\); here \(\prod_{i=1}^m(1-\alpha_iX^h)^{n_i}\) is the denominator of the rational function and the coefficients \(A_i\) are polynomials of degree \(n_i-1\). In the quotient argument one considers \(p\)-adic analytic continuations of the given generalised power sums, thus, in effect, one looks at the problem modulo \(p^n\) for sufficiently large \(n\) and a suitable infinite collection of primes \(p\), namely those for which the roots \(\alpha_i\) (and \(\beta_i\)) are units in \(\mathbb Q_p\). One then needs a new rationality criterion (dual to that employed by Dwork in his proof of the Weil conjectures) to complete the argument.

We had found that no obvious form of these methods worked for the root problem. Recall that the task here is to show there is a generalised power sum \(c\) whose \(k\)-th power is \(a\). The new idea starts from the recognition that reduction mod \(p\) of a unit yields an element of order dividing \(p-1\). I now paraphrase the author: Accordingly, one replaces the roots \(\alpha_i\) by suitable \(p-1\)-th roots of unity, where \(p=1\pmod k\). If, contrary to the conjecture, there is no \(k\)-th root \(c\) as expected then the resulting algebraic integers \(a(h)\) are not all \(k\)-th powers in the relevant cyclotomic extension of the field of definition \(k\) (this is done by appealing to the Lang-Weil theorem for the number of points on varieties over finite fields). It follows that their reduction modulo prime ideals also is is not always a \(k\)-th power in the residue class field. However for suitable \(h\) the effect of appropriate such reductions is to replace the roots of unity by the \(\alpha_i^h\). That contradicts the presumption that each original \(a(h)\) was a \(k\)-th power.

Of course, this summary glosses over several nontrivial technical difficulties.

Conversely, suppose that \((a_h)_{h\geq 0}\) and \((b_h)_{h\geq 0}\) are sequences of rational integers so that both \(\sum a_hX^h\) and \(\sum b_hX^h\) represent rational functions. Pisot conjectured that if \(b_h \bigm |a_h\) for \(h=0\), \(1\), \(\ldots \) also the Hadamard quotient \(\sum (a_h/b_h)X^h\) represents a rational function; respectively, if, say, \(a_h=c_h^3\), \(h=0\), \(1\), \(\ldots \) for some sequence \(c_h\) of rational integers then also the Hadamard cube root \(\sum c_hX^h\) is rational. This paper proves the natural generalisation of Pisot’s root conjecture.

In proving the quotient result [see ‘Solution de la conjecture de Pisot sur le quotient de Hadamard de deux fractions rationnelles’, A. J. van der Poorten, C. R. Acad. Sci., Paris, Sér. I 306, 97-102 (1988; Zbl 0635.10007)] and subsequent related work, the present reviewer noted inter alia that the integrality conditions on the sequences may be replaced by the requirement that the sequences belong to a ring finitely generated over \(\mathbb Z\) and that one readily reduces that case to that of \(S\)-integers in some algebraic number field. In that context the Hadamard \(k\)-th root conjecture asks only that the \(a_h\) all be \(k\)-th powers in some field \(\mathbf F\) finitely generated over \(\mathbb Q\). Indeed, given that, in the present paper it is proved that there then is a sequence \((c_h)\) with \(c^k_h=a_h\) for \(h=0\), \(1\), \(\ldots \) so that \(\sum c_hX^h\) is rational. Strikingly, the author’s methods involve ideas noticeably different from those applied in the proof of the quotient theorem.

Of course it is well known that if \(\sum a_hX^h\) is rational then the coefficients \(a_h=a(h)\) are given by a generalised power sum \(\sum_{i=1}^m A_i(h)\alpha_i^h\); here \(\prod_{i=1}^m(1-\alpha_iX^h)^{n_i}\) is the denominator of the rational function and the coefficients \(A_i\) are polynomials of degree \(n_i-1\). In the quotient argument one considers \(p\)-adic analytic continuations of the given generalised power sums, thus, in effect, one looks at the problem modulo \(p^n\) for sufficiently large \(n\) and a suitable infinite collection of primes \(p\), namely those for which the roots \(\alpha_i\) (and \(\beta_i\)) are units in \(\mathbb Q_p\). One then needs a new rationality criterion (dual to that employed by Dwork in his proof of the Weil conjectures) to complete the argument.

We had found that no obvious form of these methods worked for the root problem. Recall that the task here is to show there is a generalised power sum \(c\) whose \(k\)-th power is \(a\). The new idea starts from the recognition that reduction mod \(p\) of a unit yields an element of order dividing \(p-1\). I now paraphrase the author: Accordingly, one replaces the roots \(\alpha_i\) by suitable \(p-1\)-th roots of unity, where \(p=1\pmod k\). If, contrary to the conjecture, there is no \(k\)-th root \(c\) as expected then the resulting algebraic integers \(a(h)\) are not all \(k\)-th powers in the relevant cyclotomic extension of the field of definition \(k\) (this is done by appealing to the Lang-Weil theorem for the number of points on varieties over finite fields). It follows that their reduction modulo prime ideals also is is not always a \(k\)-th power in the residue class field. However for suitable \(h\) the effect of appropriate such reductions is to replace the roots of unity by the \(\alpha_i^h\). That contradicts the presumption that each original \(a(h)\) was a \(k\)-th power.

Of course, this summary glosses over several nontrivial technical difficulties.

Reviewer: A.J.van der Poorten (North Ryde)