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Problem of the Week 1208

A Little Path

For a polynomial f = a(n) x^n + ... a(1) x + a(0) with real coefficents, let P(f) be the path that starts at the origin, takes a step in the x-direction of length a(n), makes a 90° left turn, makes a second step of length a(n − 1), and so on, proceeding counterclockwise until the final step of length a(0).

Suppose that f is divisible by x² + 1. Prove that P(f) ends up at the origin.

Extra credit: Suppose P(f) ends up at the origin. Prove that f is divisible by x² + 1.

Extra credit (unsolved): For which polynomials f is P(f) a simple closed polygon, i.e., no overlap in the edges?

Source: Dan Kalman and Mark Verdi, Polynomials with closed Lill paths, Mathematics Magazine 88 (2015) 3-10. The paths are called Lill paths, hence the title of the problem.

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12 May 2015