Generalized moments, composition of polynomials and Bernstein classes (Q2782557)

Let \(P(x)\) be a polynomial of degree \(d\) and introduce for another polynomial \(q(x)\) of degree \(m-1\) and two real numbers the generalized moments NEWLINE\[NEWLINE m_k(q,a,b)=\int_a^b P^k(t)q(t) dt\;(k=0,1,2,\ldots),\quad m_{-1}=P(a)-P(b). NEWLINE\]NEWLINE The problem is now to give conditions in order to let \(m_k(q,a,b)\) vanish for all \(k=-1,0,1,\ldots\) NEWLINENEWLINENEWLINEThe so-called natural sufficient condition (also called composition condition): Let \(P\) and \(Q=\int q\) be representable as \(P(x)={\widetilde P}(W(x)), Q(x)={\widetilde Q}(W(x))\) with polynomials \({\widetilde P}, {\widetilde Q}\) and \(W\), where \(W(a)=W(b)\) is in many special cases also necessary. NEWLINENEWLINENEWLINEAn important role is played by Bernstein classes and by the number of zeros of iterated integrals of Puiseux series. The main result of the paper is then: For a given \(P\) of degree \(d\) and a given degree \(m-1\) of \(q\), there are finitely many points \(s_1,s_2,\ldots,s_M\), such that for any \(a,b\) with \(P(a)=P(b)\not= s_1,\ldots,s_M\), the vanishing of \(m_0(q,a,b),\ldots,m_N(q,a,b)\) implies the composition condition. Here \(M\) and \(N\) depend on \(d, m\) only.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].