ATRACTOR

The first step of Lill's method (cf. [2]) associates to the polynomial \[f(x) = a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1 x + a_0\] a path in the plane made up of horizontal and vertical line segments, whose lengths are determined by the coefficients of \(f\). More precisely, we begin by drawing a horizontal line segment \(\alpha_n\), of length \(|a_n|\), starting at the point \(O = (0,0)\) of the Cartesian plane, directed to the right if \(a_n > 0\) and to the left if \(a_n < 0\). Let \(P_n\) be the end point of \(\alpha_n\). If \(a_{n-1} \neq 0\), we draw another line segment \(\alpha_{n-1}\), starting at \(P_n\) but this time vertical, measuring \(|a_{n-1}|\), and directed upwards if \(a_{n-1} > 0\) and downwards if \(a_{n-1} < 0\). Let us denote by \(P_{n-1}\) the end point of \(\alpha_{n-1}\). If \(a_{n-1} = 0\), this segment \(\alpha_{n-1}\) reduces to the point \(P_n\). The following figure shows these two steps for the polynomials \(2x + 1\), \(x - 2\) and \(x\).

If \(n \geq 2\) and \(a_{n-2} \neq 0\), we continue the previous construction by drawing another horizontal segment, say \(\alpha_{n-2}\), juxtaposed to the endpoint of the last segment drawn and measuring \(|a_{n-2}|\). The direction of this new segment depends, as before, on the sign of \(a_{n-2}\): to the left if \(a_{n-2} > 0\), to the right if \(a_{n-2} < 0\). In general, the directions of the segments comply with the following rule: the curve \(\alpha\) that is drawn, composed of juxtaposed straight line segments, is oriented counterclockwise if the coefficient is positive, and clockwise if the coefficient is negative. More precisely, if the coefficients of the polynomial of degree \(n\) are all positive, then the segments corresponding to the terms of degree \(n-k\), with even \(k\), in the polynomial, are horizontal, having alternate directions and starting towards the right; and the segments associated with the terms of degree \(n-k\), with odd \(k\), in the polynomial, are vertical, also in alternate directions and starting upwards. If the polynomial has a negative coefficient, the direction of the corresponding segment is the opposite of that described; if the coefficient is zero, the segment reduces to a point. In the following figure, this construction is done for the polynomials \(x^2 - 5x + 6\), \(x^2 - 5x\) and \(x^2 + 6\).

The construction of this plane representation of the polynomial \(f\) ends when we run out of coefficients of the polynomial, having obtained a curve \(\alpha\) formed by \(\ell\) straight line segments, where \(1 \leq \ell \leq n+1\), which starts at \(O\) and ends at a point that we will designate it by \(T\). The following figure shows the curve \(\alpha\) for the polynomials \(3x^4 + x^3 + 2x^2 - 5x + 7\), \(3x^4 + 2x^2 - 5x + 7\), \(3x^4 -5x + 7\) and \(3x^4 + 2x^2 + 7\).

Next page