Irreducibility results for separated variables equations
Let K be a field of characteristic 0, \(f_ 1,...,f_{\ell}\in K(x)\) be nonconstant rational functions. Write \(f_ i=h_{i1}/h_{i2}\) where \(h_{i1},h_{i2}\in K[x]\) are coprime. Denote by \(V(f_ 1(X_ 1)+...+f_ 1(X_ 1))\) the algebraic set in affine \(\ell\)-space \({\mathbb{A}}_ K^{\ell}\) defined by the polynomial \(\prod^{\ell}_{i=1}h_{i2}(X_ i)\sum^{\ell}_{i=1}f_ i(X_ i).\) The main theorem states: If \(\ell \geq 3\) then \(V(f_ 1(X_ 1)+...+f_{\ell}(X_{\ell}))\) is irreducible. This was proved by \textit{A. Schinzel} - even for arbitrary K, but only for polynomials - [cf. Pac. J. Math. 118, 531-563 (1985; Zbl 0571.12011)]. The author gives a short proof of the main theorem using Galois-theoretic methods and results of one of his earlier papers [Ill. J. Math. 17, 128-146 (1973; Zbl 0266.14013)]. He also indicates how these results can help to shorten arguments of Schinzel in the case of positive characteristic. Now let \(\ell =2\) and \(K={\mathbb{C}}\). For positive integers n,m let \({\mathcal R}(n,m)\) denote the set of ordered pairs (h,g) of rational functions in \({\mathbb{C}}(x)\) of respective degrees n and m. To exclude trivial cases the author introduces the concept of newly reducible pairs; if h and g are polynomials then this implies that they have the same degree. - The existence of a newly reducible pair \((h,g)\in {\mathcal R}(n,m)\) is equivalent to a pure group theoretic existence problem (theorem 2.3); as a consequence one gets, that for each \(n>3\) there exists m and newly reducible \((h,g)\in {\mathcal R}(n,m)\). The existence problem for polynomials is open; if h in addition has the property that there are no proper subfields between \({\mathbb{C}}(x)\) and \({\mathbb{C}}(h(x))\) then the existence of newly reducible pairs of polynomials in \({\mathcal R}(n,m)\) implies \(n\in \{7,11,13,15,21,31\}\) [by using the classification of finite simple groups; see the author, ``Rigidity and applications of the classification of finite simple groups to monodromy'', Part II (Preprint)] and for every such n there exists (h,g) (cf. loc. cit.). The (n,m)-problem might helps to solve the existence problem mentioned above: Call a pair \((h',g')\) of polynomials in \({\mathcal R}(n,m)\) hereditarily irreducible if \(V(h'(h_ 1)-g'(g_ 1))\) is irreducible for each pair of nonconstant polynomials \((h_ 1,g_ 1)\). The (n,m) problem is the following: Does there exist an hereditarily irreducible \((h',g\}\) in \({\mathcal R}(n,m)?\) For \(n=m=2\) the answer is negative. The (2,3)-problem has an affirmative solution if there exists a group with certain properties. In the description of this group a parameter \(k=\gcd (n\cdot \deg (g_ 1),m\cdot \deg (h_ 1))/1cm(n,m)\) occurs. The author shows that in case \(k=1,2\) there is no group with the required properties. The cases \(k\geq 3\) remain open.
irreducibility of sets defined by multivariate polynomials
scientific article; zbMATH DE number 4053762
scientific article