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Algebraic Geometry for Non-Experts

What can it be used for?

Many problems can be translated into polynomial equations. Algebraic geometry is the study of the solutions of such equations.

The wavelet transformation uses a central theorem in algebraic geometry called Bezout's Theorem. Netscape's browser has a plug-in for image compression based on wavelets.

Another example is Groebner basis, which is a method for finding certain bases for ideals. Groebner basis have many applications. For example, the problem of adjusting the joints of a robot to bring the gripper arm from point A to point B can be solved using Groebner basis1. Groebner basis are also of interest in connection with artificial intelligence where the method can be used for automated theorem proving in geometry2. Groebner basis also have uses in image processing3. In order to draw three-dimensional objects, one has to work with projections, and this can be done in what is called a projective space.

It is possible to define addition of points on an elliptic curve in similar way that you add regular numbers. Instead of adding numbers you add points on the curve. This arithmetic has been used successfully in coding theory and cryptography. It is assumed, but not proved, that cryptosystems based on such arithmetic are more secure than cryptosystems based on conventional arithmetic.

References

- J. Baillieul et al., Robotics, Proceedings of Symposia in Applied Mathematics 41, American Mathematical Society 1990.

- W.T. Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, in Automated Theorem Proving: After 25 Years, edited by W. Blesoe and D. Loveland, Contemporary Mathematics 29, American Mathematical Society 1983; 213-234.

- J. Foley, A. van Damm, S. Feiner, and J. Hughes, Computer Graphics: Principles and Practice, Second Edition, Addison-Wesley 1990.