Systems of algebraic equations
We are now able to discuss the advantages of our algorithm.
The first of these is that it is as quick as we can reasonably hope : if we work with n polynomials in n variables of degree d, the general number of zeros is dn (Bezout theorem). Let N=dn. It is easy to see that the number of operations of our algorithm is less than O(N4), not counting, of course, the time needed to solve a polynomial in one variable of degree N. The longest part of the algorithm is the computation of this polynomial by interpolation.
The algorithm gives the multiplicity of each root. It permits to distinguish zeros on a particular hyperplane from the others, especially the zeros at infinity. It gives the rational relations between the coordinates of the solutions.
If working in a language which permits to compute in arbitrary algebraic extensions of the field and to factor polynomials, the algorithm permits to put together the conjugate solutions. In particular it gives the rational ones.
Finally, it works on every field, the rational one, as well as the real, the complex or finite fields.
KeywordsFinite Field Algebraic Closure Common Zero Linear Factor Classical Classis
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