##
**The rank of a Hardy field.**
*(English)*
Zbl 0536.12015

The author continues the study of Hardy fields initiated in [J. Math. Anal. Appl. 93, 297-311 (1983; Zbl 0518.12014)]. A Hardy field is a field of germs at \(+\infty\) of real valued functions on positive half-lines that is closed under differentiation. The fundamental role in the description of such fields plays the existence of a canonical valuation of a Hardy field. It is also essentially that the functions from such fields are ultimately (i.e. for sufficiently large \(x\in {\mathbb{R}})\) of constant sign. The definitions and properties of the canonical valuation \(\nu\) and the relations of equivalence and comparability are repeated in the paper. The relation of comparability is introduced in order to estimate a rate of change of the functions from a Hardy field when x tends to \(+\infty\). For example, if two functions f and g from a Hardy field are infinitely increasing, then either \(f\leq g\) or \(g\leq f\) (ultimately), and if \(f\leq g\) then f and g are comparable if and only if there is a positive integer N such that \(g\leq f^ N.\)

The author defines the rank of a Hardy field F to be the number of comparability classes of F, denoting this rk F. He proves \(rk F=card \psi,\) where \(\psi =\{\nu(u'/u):u\in F^{\times},\quad \nu(u)\neq 0\}.\) Most Hardy fields arising in practice have finite rank as it follows from Proposition 5: Let \(k\subset K\) be Hardy fields. If \(t_ 1,...,t_ n\) are algebraically dependent over k, then there are integers \(a_ 1,...,a_ n\), not all zero, such that \(\nu(t_ 1^{a_ 1}...t_ n^{a_ n})\in \nu(k^{\times}).\) In particular, if r is the transcendence degree \(r=\deg tr K/k,\) then there are at most r comparability classes of K that do not have representatives in k, so \(rk K\leq rk k+r.\)

With the help of the comparability relation it is possible to try about the solubility of some differential equations. For example, Proposition 6: Let k be a Hardy field, let \(f\in k\) be infinitely increasing, and suppose k has comparability classes smaller then that of f. Then there exists \(g\in k\) such that \(g\sim \log f.\)

It is also possible to estimate the order of growth of solutions of ordinary differential equations. Such appraisals are given in theorems 2 and 3 (where \(\ell_ n(x)=...\log x_{n\quad times}\) and \(e_ n(x)=...\exp x,_{n\quad times}\) also \(e_ n(x)=\ell_{-n}(x)).\)

Theorem 2: Let k be a Hardy field of finite positive rank n. Let w be an infinitely increasing element of the largest comparability class of k. If k contains an element \(u_ 0\sim x\), then there exist nonnegative integers r, s, with \(r+s+1\leq n,\) and a positive real number A such that k contains elements \(u_ 0,u_ 1,...,u_ r\), with each \(u_ i\sim \ell_ i(x)\), such that \(u_ r\) is in the smallest comparability class of k, and such that \(w<e_ s(x^ A).\) If k contains no \(u_ 0\sim x\), then for some positive integer \(s\leq n\), k contains an element \(u\sim \ell_{s-1}(w)\) that is contained in the smallest comparability class of k, and we have u’/\(u\sim C\), a positive real number, so for any real \(\epsilon>0\) we have \(e^{(C-\epsilon)x}<u<e^{(C+\epsilon)x}\) and \(e_ s((C-\epsilon)x)<w<e_ s((C+\epsilon)x).\)

Theorem 3: Let \(k\subset K\) be Hardy fields, where K contains at most n elements that are mutually incomparable and incomparable to any element of k. Also suppose k contains a smallest comparability class. Then K contains a smallest comparability class and there are integers r,\(s\geq 0\) satisfying \(r+s\leq n\) such that the smallest comparability class of K contains an element \(\sim \ell_ r(f)\) for any infinitely increasing f in the smallest comparability class of k, and for any given \(t\in K\) there exists \(g\in k\) such that \(t<e_ s(g).\)

M. Boshernitzan obtained results in this direction by different methods [see J. Anal. Math. 39, 235-255 (1981) and ibid. 41, 130-167 (1982)].

In the paper the notion of the rational rank of a Hardy field and the original proof of differential transcendence over \({\mathbb{R}}\) of the gamma function \(\Gamma\) (x) and the Riemann zeta function \(\zeta\) (x) are considered. It is proved that the rank of \({\mathbb{R}}<\Gamma(x)>\) is equal to 3, and for \({\mathbb{R}}<\zeta(x)>\) it is 1.

The paper is written rather in an algebraic then analytic taste.

The author defines the rank of a Hardy field F to be the number of comparability classes of F, denoting this rk F. He proves \(rk F=card \psi,\) where \(\psi =\{\nu(u'/u):u\in F^{\times},\quad \nu(u)\neq 0\}.\) Most Hardy fields arising in practice have finite rank as it follows from Proposition 5: Let \(k\subset K\) be Hardy fields. If \(t_ 1,...,t_ n\) are algebraically dependent over k, then there are integers \(a_ 1,...,a_ n\), not all zero, such that \(\nu(t_ 1^{a_ 1}...t_ n^{a_ n})\in \nu(k^{\times}).\) In particular, if r is the transcendence degree \(r=\deg tr K/k,\) then there are at most r comparability classes of K that do not have representatives in k, so \(rk K\leq rk k+r.\)

With the help of the comparability relation it is possible to try about the solubility of some differential equations. For example, Proposition 6: Let k be a Hardy field, let \(f\in k\) be infinitely increasing, and suppose k has comparability classes smaller then that of f. Then there exists \(g\in k\) such that \(g\sim \log f.\)

It is also possible to estimate the order of growth of solutions of ordinary differential equations. Such appraisals are given in theorems 2 and 3 (where \(\ell_ n(x)=...\log x_{n\quad times}\) and \(e_ n(x)=...\exp x,_{n\quad times}\) also \(e_ n(x)=\ell_{-n}(x)).\)

Theorem 2: Let k be a Hardy field of finite positive rank n. Let w be an infinitely increasing element of the largest comparability class of k. If k contains an element \(u_ 0\sim x\), then there exist nonnegative integers r, s, with \(r+s+1\leq n,\) and a positive real number A such that k contains elements \(u_ 0,u_ 1,...,u_ r\), with each \(u_ i\sim \ell_ i(x)\), such that \(u_ r\) is in the smallest comparability class of k, and such that \(w<e_ s(x^ A).\) If k contains no \(u_ 0\sim x\), then for some positive integer \(s\leq n\), k contains an element \(u\sim \ell_{s-1}(w)\) that is contained in the smallest comparability class of k, and we have u’/\(u\sim C\), a positive real number, so for any real \(\epsilon>0\) we have \(e^{(C-\epsilon)x}<u<e^{(C+\epsilon)x}\) and \(e_ s((C-\epsilon)x)<w<e_ s((C+\epsilon)x).\)

Theorem 3: Let \(k\subset K\) be Hardy fields, where K contains at most n elements that are mutually incomparable and incomparable to any element of k. Also suppose k contains a smallest comparability class. Then K contains a smallest comparability class and there are integers r,\(s\geq 0\) satisfying \(r+s\leq n\) such that the smallest comparability class of K contains an element \(\sim \ell_ r(f)\) for any infinitely increasing f in the smallest comparability class of k, and for any given \(t\in K\) there exists \(g\in k\) such that \(t<e_ s(g).\)

M. Boshernitzan obtained results in this direction by different methods [see J. Anal. Math. 39, 235-255 (1981) and ibid. 41, 130-167 (1982)].

In the paper the notion of the rational rank of a Hardy field and the original proof of differential transcendence over \({\mathbb{R}}\) of the gamma function \(\Gamma\) (x) and the Riemann zeta function \(\zeta\) (x) are considered. It is proved that the rank of \({\mathbb{R}}<\Gamma(x)>\) is equal to 3, and for \({\mathbb{R}}<\zeta(x)>\) it is 1.

The paper is written rather in an algebraic then analytic taste.

Reviewer: N.V.Grigorenko

### MSC:

12H05 | Differential algebra |

12H20 | Abstract differential equations |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |

12F20 | Transcendental field extensions |

26A12 | Rate of growth of functions, orders of infinity, slowly varying functions |

11J81 | Transcendence (general theory) |

### Keywords:

rank of Hardy field; differential transcendence of gamma function; differential transcendence of Riemann zeta function; Hardy fields; canonical valuation; transcendence degree; comparability relation; order of growth of solutions of ordinary differential equations; comparability class### Citations:

Zbl 0518.12014
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\textit{M. Rosenlicht}, Trans. Am. Math. Soc. 280, 659--671 (1983; Zbl 0536.12015)

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### References:

[1] | Steven B. Bank and Robert P. Kaufman, A note on Hölder’s theorem concerning the gamma function, Math. Ann. 232 (1978), no. 2, 115 – 120. · Zbl 0354.33001 |

[2] | Michael Boshernitzan, New ”orders of infinity”, J. Analyse Math. 41 (1982), 130 – 167. · Zbl 0539.26003 |

[3] | Michael Boshernitzan, ”Orders of infinity” generated by difference equations, Amer. J. Math. 106 (1984), no. 5, 1067 – 1089. , https://doi.org/10.2307/2374273 Michael Boshernitzan, Discrete ”orders of infinity”, Amer. J. Math. 106 (1984), no. 5, 1147 – 1198. · Zbl 0602.26003 |

[4] | N. Bourbaki, Fonctions d’une variable réele. Chapitre VII (”La Fonction Gamma”), 2nd ed., Hermann, Paris, 1961. |

[5] | O. Hölder, Über die Eigenschaft der \( \Gamma \)-Funktion, keiner algebraischen Differentialgleichung zu genügen, Math. Ann. 28 (1887), 1-13. · JFM 18.0440.02 |

[6] | F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. · Zbl 0303.41035 |

[7] | Maxwell Rosenlicht, Differential valuations, Pacific J. Math. 86 (1980), no. 1, 301 – 319. · Zbl 0401.12024 |

[8] | Maxwell Rosenlicht, On the value group of a differential valuation, Amer. J. Math. 101 (1979), no. 1, 258 – 266. · Zbl 0411.12021 |

[9] | Maxwell Rosenlicht, Hardy fields, J. Math. Anal. Appl. 93 (1983), no. 2, 297 – 311. · Zbl 0518.12014 |

[10] | M. Singer, Asymptotic behavior of solutions of differential equations and Hardy fields, preliminary report, SUNY at Stony Brook, 1976 (unpublished). |

[11] | L. van den Dries, Bounding the rate of growth of solutions of algebraic differential equations and exponential equations in Hardy fields (to appear). |

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