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372 where the a j,i 1 ,...,i n are indeterminates. If we write g 1 f 1 + · · · + g m f m as a polynomial in the variables x 1 , . . . , x n , then all the coefficients must vanish, save the constant term which must equal 1. Thus we get a system of linear equations in the indeterminates a j,i 1 ,...,i n . The solvability of systems of linear equations is well-known (with good computer implementations). Thus we can decide if there is a solution with deg g j 100. Of course it is possible that 100 was too small a guess, and we may have to repeat the process with larger and larger degree bounds. Will this ever end? The answer is given by the following result, which was proved only recently. Effective Nullstellensatz. Let f 1 , . . . , f m be polyno- mials of degree less than or equal to d in n variables, 3, n 2. If they have no common zero, where d IV. Branches of Mathematics with many nice pictures and reproductions. A Scrap- book of Complex Curve Theory (American Mathemat- ical Society, Providence, RI, 2003), by C. H. Clemens, and Complex Algebraic Curves (Cambridge University Press, Cambridge, 1992), by F. Kirwan, also start at an easily accessible level, but then delve more quickly into advanced subjects. The best introduction to the techniques of algebraic geometry is Undergraduate Algebraic Geometry (Cam- bridge University Press, Cambridge, 1988), by M. Reid. For those wishing for a general overview, An Invitation to Algebraic Geometry (Springer, New York, 2000), by K. E. Smith, L. Kahanpää, P. Kekäläinen, and W. Traves, is a good choice, while Algebraic Geometry (Springer, New York, 1995), by J. Harris, and Basic Algebraic Geometry, volumes I and II (Springer, New York, 1994), by I. R. Shafarevich, are suitable for more systematic readings.