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Computing the roots of sparse high–degree polynomials that arise from the study of random simplicial complexes

  • Author(s): Farouki, RT
  • Strom, JA
  • et al.

The problem of computing the roots of a particular sequence of sparse polynomials p (t) is considered. Each instance p (t) incorporates only the n + 1 monomial terms t,t2,t4,…,t2n associated with the binomial coefficients of order n and alternating signs. It is shown that p (t) has (in addition to the obvious roots t = 0 and 1) precisely n − 1 simple roots on the interval (0,1) with no roots greater than 1, and a recursion relating p (t) and p (t) is used to show that they possess interlaced roots. Closed–form expressions for the Bernstein coefficients of p (t) on [0,1] are derived and employed to compute the roots in double–precision arithmetic. Despite the severe variation of the graph of p (t), and tight clustering of roots near t = 1, it is shown that for n ≤ 10, the roots on (0,1) are remarkably well–conditioned and can be computed to high accuracy using both the power and Bernstein forms. n n n n n+ 1 n n

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