Factorization in \(K[X^ 2,X^ 3]\) (Q811394)

Let \(K\) be a field and \(R=K[X^ 2,X^ 3]\). In this paper, we study lengths of factorizations in \(R\). For any atomic domain \(D\), define \(\rho(D)=\sup\{m/n\mid x_ 1\cdots x_ m= y_ 1\cdots y_ n\), \(y_ j\in D\) irreducible\} and \(\Phi(n)= |\{m\mid x_ 1\cdots x_ m= y_ 1\cdots y_ n\), \(x_ i, y_ j\in D\) irreducible\}\(|\). We show that \(\rho(R)=(D(K)+ 2)\), where \(D(K)\) is the Davenport constant of \(K\) as an additive abelian group. Hence \(\rho(R)\) is finite if and only if \(K\) is finite. If \(K\) is finite, we also show that \(\lim_{n\to\infty} \Phi(n)/n=(\rho(R)^ 2-1)/\rho(R)\).