The Billaud Conjecture for vertical bar Sigma vertical bar=4, and Beyond

The Billaud Conjecture, first stated in 1993, is a fundamental problem on finite words and their heirs, i.e., the words obtained by a projection deleting a single letter. The conjecture states that every morphically primitive word, i.e., a word which is not a fixed point of any non-identity morphism, has at least one morphically primitive heir. In this paper we give the proof of the Conjecture for alphabet size 4, and discuss the potential for generalising our reasoning to larger alphabets. We briefly discuss how other language-theoretic tools relate to the Conjecture, and their suitability for potential generalisations.