ATRACTOR

With the interactive modules produced by Attractor, available here, we can consider a polynomial, apply Lill's method to check if it has real zeros, and even change the coefficients of the polynomial to analyze how real zeros and the corresponding Lill polygons vary.

This is what the following figure shows for the parameterized family of polynomials given by \(f_\lambda(x) = x^2 - 5x + \lambda\), with \(\lambda \in \{0, 4, 6, 25/4\}\).

This interactive component on Lill's method can also be used to answer some questions suggested by the method, which the reader is invited to explore. For example:

1. Given a polynomial \(f\) of degree greater than or equal to 2 and a path \(\beta_f\) that determines a real zero\(x_0\) of \(f\) through Lill's method, then \(\beta_f\) is the polygonal curve \(\alpha_g\) of the polynomial \(g = f/(x - x_0)\) if properly repositioned (that is, with the figure rotated so that the initial segment of \(\beta_f\) is horizontal) and then rescaled by an homothety.

This property is illustrated in the following figure for the polynomial \(x^3 - 2x^2 - 5x + 6.\)

The first image of the previous figure represents, in different colors (magenta, orange and brown), three \(\beta\) paths that connect \(O\) to \(T\) and have initial slopes relative to the first black line segment (horizontal) equal to \(k=2\), \(k=-1\) and \(k=-3\), respectively. And therefore, from what we stated earlier, the polynomial has three real zeros, \(-2\), \(1\) e \(3\). In the middle image, obtained by choosing \(k=2\), we proceed with a construction of \(\beta\) curves similar to the previous one, but now with initial slopes relative to the first magenta segment instead of the first black horizontal segment. Thus, we find two choices of slopes that generate Lill polygons, \(k=-1\) and \(k=-3\), precisely the two values of \(k\) that were not used to pass from the first image to the second. Somehow, it's as if the magenta curve represents the polynomial of the second degree which is the quotient of the initial polynomial by \(x+2\), and we can repeat the construction with this new curve to get the other real zeros of the initial polynomial; and, therefore, it is as if this geometric construction allows the visualization of the division of the initial polynomial by \(x+2\). To make this iteration of the method easier to understand, we represent in the third image the result of a rotation that placed the first magenta segment horizontally.

2. The \(\alpha\) curves of the polynomials shown in the following figure are all closed, that is, \(T = O\). What do these polynomials have in common?

The answer (they are all divisible by \(x^2 + 1\)) is somewhat surprising, but in fact this is a particular case of a general result: the polygonal curve \(\alpha\) of a polynomial \(f\) is closed if and only if \(x^2 + 1\) divides \(f\). A proof of this property can be found in [1].

3. The configuration of the \(\alpha\) and \(\beta\) curves admits some generalizations. For example, we can change the \(90^\circ\) angle between the line segments \(\alpha_j\) that form the curve \(\alpha\); or change the \(90^\circ\) angle at which the \(\beta\) curve segments are reflected in the vertical or horizontal directions. In [1], the reader finds additional information about these more general approaches.