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Differential equations over polynomially bounded o-minimal structures. (English) Zbl 1007.03039

Let \(\mathbb R\) denote the ordered field of the reals. An expansion \(\mathfrak R\) of \(\mathbb R\) is said to be o-minimal if every set of reals definable in \(\mathfrak R\) is a finite union of points and intervals. It is called polynomially bounded if for every definable unary function \(f\) there is a natural number \(d\) such that \(f(t)\leq t^d\), for all sufficiently large \(t\). C. Miller (1994) discovered that any o-minimal expansion of \(\mathbb R\) is either polynomially bounded or defines the exponential function. It is easy to show that \(\mathbb R\) itself is polynomially bounded and o-minimal; at present examples of large classes of maps that generate polynomially bounded o-minimal expansions of \(\mathbb R\) are known. P. Speissegger (1999) defined \({\mathcal P}(\mathfrak R)\), the Pfaffian closure of \(\mathfrak R\), as the smallest expansion of \(\mathfrak R\) that is closed under taking Rolle leaves of definable \(C^1\) forms. He proved that \({\mathcal P}(\mathfrak R)\) is o-minimal if \(\mathfrak R\) is o-minimal. Note that every presently known o-minimal structure over \(\mathbb R\) is a reduct of \({\mathcal P}(\mathfrak R)\), for some polynomially bounded o-minimal \(\mathfrak R\).
Let \(\mathfrak R\) be a polynomially bounded o-minimal expansion of \(\mathbb R\). The authors investigate the asymptotic behavior at \(+\infty\) of non-oscillatory solutions to differential equations \(y'=g(t,y)\), where \(g:\mathbb R^{1+l}\to\mathbb R^l\) is definable in \(\mathfrak R\). To this end, they develop the o-minimal analog of Rosenlicht’s Hardy field theory [M. Rosenlicht, Trans. Am. Math. Soc. 280, 659-671 (1983; Zbl 0536.12015), ibid. 299, 261-272 (1987; Zbl 0619.34057)]. Among other results, they show that \({\mathcal P}(\mathfrak R)\) is levelled. This means that any infinitely increasing function definable in \({\mathcal P}(\mathfrak R)\) has level in the sense of Rosenlicht; the level is a certain integer characterising growth of the function in comparison with the functions \(e_n\), \(n\in \mathbb Z\), where \(e_0(t)=t\), \(e_{n+1}(t)=\exp(e_n(t))\), and \(e_{n-1}(t)=\log(e_n(t))\). In particular, \({\mathcal P}(\mathfrak R)\) is exponentially bounded, that is, for each unary function \(f\) definable in \({\mathcal P}(\mathfrak R)\) there exists a positive \(n\) such that ultimately \(f(t)\leq e_n(t)\). Some of the results hold in o-minimal expansions of arbitrary ordered fields.

MSC:

03C64 Model theory of ordered structures; o-minimality
34E05 Asymptotic expansions of solutions to ordinary differential equations
26A12 Rate of growth of functions, orders of infinity, slowly varying functions
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