USNA Mathematics Department seminar

Title:

Abstract: Computer hardware has a slightly different conception of mathematics than that of the mathematician. My computer, for example, thinks that the natural numbers end at 2^64 - 1, and believes it has a terminating decimal representation of the square root of 2.

Computer algebra is all about bringing the computer's idea of mathematics in line with the mathematician's view. The most familiar example of this is probably the arbitrarily-large-integer arithmetic of a computer algebra system like Maple or Mathematica. But computer algebra goes beyond computing with numbers, allowing us to compute in the exact mathematical sense with symbolic objects, like polynomials and integrals, and with geometric objects.

I'll talk about how the computer can be made to compute with the objects of elementary real algebra: parabolas, ellipsoids, half-spaces, etc. For example:

• Does the disk centered at the origin with radius 1/2 intersect with the curve defined by 3y² - x³ + 4x - 2? No.
• For what radii do they intersect? r < 162/89 - 63/178 sqrt(21)