Recent zbMATH articles in MSC 37F75https://www.zbmath.org/atom/cc/37F752021-03-30T15:24:00+00:00WerkzeugAnalytic deformations of pencils and integrable one-forms having a first integral.https://www.zbmath.org/1455.370442021-03-30T15:24:00+00:00"Scárdua, Bruno"https://www.zbmath.org/authors/?q=ai:scardua.bruno|scardua.bruno-c-azevedoThe paper addresses the problem of persistence of a first integral for a holomorphic integrable one-form in a neighborhood of a singular point.
The author considers integrable analytic deformations of codimension-one holomorphic foliations. Let \(\Omega(x,t)\) be a holomorphic one-form defined in a connected product neighborhood \(U \times D\subset \mathbb{C}^n\times\mathbb{C}\), of the origin of \(\mathbb{C}^{n+1}\), where \(D\subset\mathbb{C}\) is a disc centered at the origin, such that \(\Omega^0:=\Omega(x,0)\) admits a first integral of holomorphic or meromorphic type. For each parameter \(t\in D\), \(\Omega^t\) denotes the restriction \(\Omega^t:=\Omega(x,t)\). The author considers the case where each \(\Omega^t\) is integrable. The family of codimension-one foliations \(\{\mathcal{F}_t:\Omega^t=0~\text{in}~U\}_{t\in(\mathbb{C},0)}\) can be seen as an analytic deformation in \(U\) of the foliation \(\mathcal{F}_0\) which, by hypothesis, admits a holomorphic (or meromorphic) first integral \(f\).
In the first part of the paper, the author studies the case where the foliation \(\mathcal{F}_0\) has a holomorphic first integral \(f\) at the origin in \(\mathbb{C}^n\) with \(n\ge 3\), under the assumption that the germ \(f\) is irreducible and that its typical fiber is simply-connected. In Theorem 1.1, the author first proves that if the codimension of the singular locus of \(\Omega^0\) is greater than or equal to \(2\), then the germ of the foliation \(\mathcal{F}_t\) also has a holomorphic first integral. Then, for the general case, that is when the codimension of the singular locus of \(\Omega^0\) is greater than or equal to \(1\), the author obtains a two-dimensional normal form for the foliation \(\mathcal{F}_t\).
The second part of the paper deals with analytic deformations \(\{\mathcal{F}_t\}_{t\in(\mathbb{C},0)}\) of a local pencil \(\mathcal{F}_0:\frac{f}{g}=\hbox{constant}\) for \(f,g\) belonging to the ring \(\mathcal{O}_n\) of germs of holomorphic functions at the origin of \(\mathbb{C}^n\). In dimension \(n=2\), the author considers \(f=x\) and \(g=y\) and assumes that the axes are invariant for each foliation \(\mathcal{F}_t\).
In higher dimension \(n\ge 3\), the author works under some generic geometric conditions on the germs \(f\) and \(g\). In both cases, the author proves the following (Theorems 1.2 and 1.3):
1) For an analytic deformation there is a multiform formal first integral of type \(\widehat F = \frac{f^{1+\widehat\lambda(t)}}{g^{1+\widehat\mu(t)}} e^{\widehat H(x,y,t)}\) with \(\widehat\lambda(t)\) and \(\widehat\mu(t)\) formal series and some properties on \(\widehat H(x,y,t)\);
2) For an integrable deformation there is a meromorphic first integration of the form \(M = \frac{f}{g} e^{P(t)+H(x,y,t)}\) with \(P(t)\) holomorphic and some additional properties on \(H(x,y,t)\).
Reviewer: Jasmin Raissy (Toulouse)A characterization of multiplicity-preserving global bifurcations of complex polynomial vector fields.https://www.zbmath.org/1455.370422021-03-30T15:24:00+00:00"Dias, Kealey"https://www.zbmath.org/authors/?q=ai:dias.kealeySummary: For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree \(d\), it is proved that any bifurcation which preserves the multiplicity of equilibrium points admits a decomposition into a finite number of elementary bifurcations, and the elementary bifurcations are characterized.