First Lower Bounds for Palindromic Length.

We study possible behaviour of the function of prefix palindromic length PPLu(n) of an infinite word u, that is, the minimal number of palindromes to which the prefix of length n of u can be decomposed. In a 2013 paper with Puzynina and Zamboni we stated the conjecture that PPLu(n) is unbounded for every infinite word u which is not ultimately periodic. Up to now, the conjecture has been proved only for some particular cases including all fixed points of morphisms and, later, Sturmian words. To give an upper bound for the palindromic length, it is in general sufficient to point out a decomposition of a given word to a given number of palindromes. Proving that such a decomposition does not exist is a trickier question. In this paper, we summarize the existing techniques which can be used for lower bounds on the palindromic length. In particular, we completely describe the prefix palindromic length of the Thue-Morse word and use appropriate numeration systems to give a lower bound for the palindromic length of some Toeplitz words.