On maps of tori (Q2455389)

The purpose of the paper is to study polynomial maps between products of generalized spheres. Given a field \(\mathbb{K}\), the \(n\)-sphere \(S^n(\mathbb{K})\subset \mathbb{K}^{n+1}\) is the zero locus of the polynomial \(x_0^2+\dots+x_n^2-1\), and the \(n\)-torus \(T^n(\mathbb{K})\) is the product \(S^1(\mathbb{K})\times\cdots\times S^1(\mathbb{K})\). The authors first characterize polynomial maps \(S^n(\mathbb{K})\to S^1(\mathbb{K})\). Based on that, a description of all polynomial maps \(S^{m_1}(\mathbb{K})\times\cdots\times S^{m_r}(\mathbb{K})\to T^n(\mathbb{K})\) is derived. The rest of the paper deals with maps of real tori of the form \(S^{2n_1-1}\times\cdots\times S^{2n_r-1}\to T^n\) and \(T^n\to S^{2n_1-1}\times\cdots\times S^{2n_r-1}\), which are induced from holomorphic maps.