ATRACTOR

Once the curve \(\alpha\) is drawn, another curve is constructed, let’s call it \(\beta\), depending on the initial choice of an angle \(\theta\):

1. The first line segment \(\beta_1\) of \(\beta\) starts at \(O\), making an angle \(\theta\) with the \(x\)-axis, and ends at a point \(Q_1\) on the vertical line that passes through \(P_n\).
2. The second line segment \(\beta_2\) from \(\beta\) starts at \(Q_1\) and makes an angle of \(90^\circ\) with \(\beta_1\). This change of angle in the direction of movement of \(\beta_1\) requires a choice whether the continuation \(\beta_2\) is made on the other side of the vertical line or on the same side as \(\beta_1\). This choice is made so that segment \(\beta_2\) intersects the horizontal line determined by the coefficient \(a_{n-1}\) of the polynomial \(f\).
3. We continue with the process described above, up to the intersection with the line that contains the last segment of curve \(\alpha\).

In general, this last point on curve \(\beta\) is not the last point \(T\) on curve \(\alpha\). It coincides with it if and only if the number \(x_0 = - \text{tangent}\, (\theta)\) is a real zero of the polynomial. That is, equation (1) has a real solution if and only if it is possible to join \(O\) to \(T\) by a curve \(\beta\) as previously defined.¹

If such a curve \(\beta\) exists, the curves \(\alpha\) and \(\beta\) form a polygon \(\mathcal{L}\), (which possibly degenerates into a segment), which we will call the Lill polygon of the polynomial \(f\). This is what happens in the examples in the following figure, which refer to the polynomials \(2x + 1\), \(x^2 - 2x + 1\), \(x^2 - 2\), \(x^2 - 5x + 6\) and in which the curves \(\alpha\) and \(\beta\) are drawn in black and magenta, respectively. Note that there are several possible curves \(\beta\), for different initial angles \(\theta\), if the polynomial equation has more than one real solution.

The following figures contain similar information about the polynomials \(x^3 - 9\),\(x^3 - 2x^2 - 5x - 2\), \(x^3 - 7x - 6\) and \(x^4 - 5x^3 + 8x^2 - 10x + 12\).

Here you will find an interactive application with a cursor with which, for each polynomial, you can vary the angle \(\theta\), see how the curve \(\beta\) changes and even search for the positions where it ends in \(T\), that is, search for the real zeros of the chosen polynomial.

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