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Cubic Equations

By ellipticcurves

Now assume that f(x,y) is cubic. The previous method no longer works. Indeed, if  we have a point P(x_0, y_0) \in C(K) and draw a line of slope \alpha \in K, then equation

f(x, \alpha (x - x_0) + y_0)=0,

which we need to solve to find the points of intersection of the line with the curve C, will be cubic:

ax^3+bx^2+cx + d=0.

After factoring out x-x_0, we are left with a quadratic equation, whose roots need not lie in K.

L. J. Mordell came up with the following way to salvage this method. Suppose that we know of two points P(x_0, y_0), Q(x_1, y_1) \in C(K). Then the line L through P and Q has slope

\alpha = \frac{y_1 - y_0}{x_1-x_0} \in K,

so we can apply the above method. But this time we know of two roots of the equation ax^3+bx^2+cx + d=0 resulting from substitution of the equation of L into the equation for C. Since the sum of solutions x_0 + x_1 + x_2 = -b/a, the third solution x_2 = -b/a - x_0 - x_1 also lies in K.

There are many “degenerate” cases of the above procedure. For example, a similar process can be followed if P=Q by using the equation of the line tangent to C at P. The slope of that line will also lie in K.

The immediate purpose of this class is to organize the above procedure into a consistent method, which can then be used to systematically find all solutions to cubic equations.

Example. Let C be the curve defined by y^2=x^3+2x+3. It contains two points: P(-1,0) and Q(1/4,15/8). The line through them has slope \alpha=\frac{3}{2} and equation y=\frac{3}{2} (x+1). Plug that into the equation of the curve to get

(\frac{3}{2}(x+1))^2 = x^3 + 2x + 3,

which simplifies to

x^3 - \frac{9}{4}x^2 + \cdots = 0.

Notice that we don’t really care what the other terms in that equation are. We conclude that (-1) + 1/4 + x_2 = 9/4, and therefore, x_2 = 3. Finally, y_2 = \frac{3}{2}(x_2+1) = 6. Thus, the third point of intersection of the line through P and Q is (3, 6). We denote this point by P*Q. Note that since the curve is symmetric with respect to the x-axis, the reflection of P*Q with respect to the x-axis will also have rational coordinates.

Bibliography: 

[1] Mordell, L. J. “On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees.” Proc. Cambridge Philos. Soc. 21, 179-192, 1922-23.

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This entry was posted on August 26, 2008 at 5:52 pm and is filed under algebraic geometry. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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