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 had a plug-in for image compression based on wavelets.
Another example is the Groebner basis, which is a method for finding specific bases for ideals. Groebner basis has 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 basis. The Groebner basis is also of interest in connection with artificial intelligence, where the method has been used for automated theorem proving in geometry. Groebner basis has also been used in image processing. 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 the addition of points on an elliptic curve in a 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.
- J. Baillieul et al., Robotics, Proceedings of Symposia in Applied Mathematics 41, American Mathematical Society 1990.