Growth of transcendental entire functions on algebraic varieties (Q1288503)

Let \(X\) be a complete intersection algebraic variety of codimension \(M>1\) in \(\mathbb{C}^{m+n}\) and \(K\) a non-pluripolar compact subset of \(X\). The authors extend the notion of \((p,q)\)-order, \((p,q)\)-\(K\)-type to transcendental entire functions \(f \in \mathcal{O}(\mathbb{C}^{m+n})\). Suppose \(f \in \mathcal{O}(X)\) has the series expansion in terms of an orthogonal polynomial basis in a Hilbert space \(\mathcal{L}^{2}(X,\mu)\), where \(\mu\) is the capacity extremal measure on \(K\). The main result proved here is a necessary and sufficient condition for such \(f\) to be of \((p,q)\)-order \(\rho (b<\rho<\infty)\) and \((p,q)\)-\(K\)-type \(\sigma\) with respect to any proximate order \(\rho(r)\).