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Let A=SUMi=0mAi xi and B=SUMi=0nBi xi be non-zero polynomials over an integral domain I, i.e.,
A, B inI[x]The Sylvester matrix of A and B is the m+n by m+n matrix
whose upper part consists of n rows of the coefficients of A and whose lower part consists of m rows of the coefficients of B (all entries not shown are zero). The resultant of A and B is the determinant of this matrix, i.e.,
Am Am-1 ... A0 ... ... ... ... Am Am-1 ... A0 Bn Bn-1 ... B0 ... ... ... ... Bn Bn-1 ... B0
Resultant(A,B) inI.
In the context of the plotting of algebraic curves, we are interested in the special case where A and B are polynomials in r variables over the integers, i.e., given
A,B inZ[x1,...,xr-1][xr]we want to find
Resultant(A,B) inZ[x1,...,xr-1].For plotting algebraic curves in two dimensions, we have to solve this problem for r=2.