Palindromic Length and Reduction of Powers
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Given a nonempty finite word v, let PL(v) be the palindromic length of v; it means the minimal number of palindromes whose concatenation is equal to v. Let v^R denote the reversal of v. Given a finite or infinite word y, let Fac(y) denote the set of all finite factors of y and let maxPL(y)=max{PL(t)| t∈ Fac(y)}. Let x be an infinite non-ultimately periodic word with maxPL(x)=k<∞ and let u∈ Fac(x) be a primitive nonempty factor such that u^5 is recurrent in x. Let Ψ(x,u)={t∈ Fac(x)| u,u^R∉Fac(t)} We construct an infinite non-ultimately periodic word x such that u^5, (u^R)^5∉Fac(x), Ψ(x,u)⊆ Fac(x), and maxPL(x)≤ 3k^3. Less formally said, we show how to reduce the powers of u and u^R in x in such a way that the palindromic length remains bounded.
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