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PolynomialIdeals[NumberOfSolutions] - compute the number of solutions over the algebraic closure
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Calling Sequence
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NumberOfSolutions(J)
NumberOfSolutions(G, tord)
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Parameters
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J
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a polynomial ideal
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G
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a Groebner basis
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J
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a monomial order
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Description
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The NumberOfSolutions command computes the number of solutions of a system over the algebraic closure of the coefficient field, including multiplicities. A zero-dimensional system has a finite number of solutions.
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This function is part of the PolynomialIdeals package, and can be used in the form NumberOfSolutions(..) only after executing the command with(PolynomialIdeals). However, it can always be accessed through the long form of the command using PolynomialIdeals[NumberOfSolutions](..).
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Examples
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Observe that the generators of J are already a Groebner basis with respect to plex(x,y). The monomials not divisible by x^2 or y^3 are
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![M := [seq(seq(x^i*y^j, j = 0 .. 2), i = 0 .. 1)]](/support/helpjp/helpview.aspx?si=8512/file02592/math131.png)
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![M := [1, y, y^2, x, x*y, x*y^2]](/support/helpjp/helpview.aspx?si=8512/file02592/math134.png)
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References
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Cox, D.; Little, J.; and O'Shea, D. Using Algebraic Geometry. New York: Springer-Verlag, 1998.
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