Problem of the Week

The graph of the circle of radius 1 consists of the solutions of the equation x² + y² - 1 = 0. This graph contains many rational points, that is, points both of whose coordinates are rational numbers; for example, (3/5, 4/5) satisfies this property.  This graph is also an example of the roots of an integral polynomial in two variables, in general, an integral polynomial in two variables is a sum of terms of the form k x^ay^b, where k is any integer and a and b are non-negative integers.

Can you find an integral polynomial in two variables with NO (or at most finitely many) rational points as roots?  Give an explicit example (and an explanation for the dearth of such points) or explain why no such graph is possible.

Note: Yes, I know the function f(x,y) = x² + y² + 1 will do the job, but that violates the spirit of the thing! Find one with a nice rich graph -- for example the graph should have no isolated points.

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ã2002 Alberto L. Delgado