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RegularChains

 Intersect
 compute the common solutions of a polynomial and a regular chain

 Calling Sequence Intersect(f, rc, R)

Parameters

 f - polynomial rc - regular chain R - polynomial ring

Description

 • The command Intersect(f, rc, R) computes the common solutions of the polynomial f and the regular chain rc in the following sense. Let V be the hypersurface defined by f, that is, the solutions of the equation f = 0 . Let W be the quasi-component of rc. Then Intersect(f, rc, R) returns regular chains such that the union of their quasi-components contains the intersection of V and W, and this union is contained in the intersection of V and the Zariski closure of W. See ConstructibleSetTools for a definition of a quasi-component.
 • When the regular chain rc has dimension zero, Intersect(f, rc, R) computes exactly the intersection of V and W. This is also the case when W is a variety (that is a closed set for Zariski topology) or when rc has dimension one and f is regular w.r.t. the saturated ideal of rc. In all other cases, Intersect(f, rc, R) computes a superset of the intersection of V and W. However this superset is very close to this intersection.
 • In summary and in broad terms, Intersect(f, rc, R) computes a sharp approximation of the intersection of V and W by means of regular chains.
 • You can use the function Intersect to solve systems of equations incrementally, that is, one equation after the other. The example below illustrates this strategy.
 • Another way of understanding the Intersect command is to observe that it specializes the solutions of rc with the constraint f = 0 .

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$

Define a ring of polynomials.

 > $\mathrm{vars}≔\left[x,y,z\right]:$$R≔\mathrm{PolynomialRing}\left(\mathrm{vars}\right):$

Define a set of equations.

 > $\mathrm{sys}≔\left[{x}^{2}+y+z-1,x+{y}^{2}+z-1,x+y+{z}^{2}-1\right]$
 ${\mathrm{sys}}{≔}\left[{{x}}^{{2}}{+}{y}{+}{z}{-}{1}{,}{{y}}^{{2}}{+}{x}{+}{z}{-}{1}{,}{{z}}^{{2}}{+}{x}{+}{y}{-}{1}\right]$ (1)

Define the empty regular chain.

 > $\mathrm{rc}≔\mathrm{Empty}\left(R\right)$
 ${\mathrm{rc}}{≔}{\mathrm{regular_chain}}$ (2)

Solve the first equation.

 > $\mathrm{dec}≔\mathrm{Intersect}\left(\mathrm{sys}\left[1\right],\mathrm{rc},R\right);$$\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}\right]$
 $\left[\left[{{x}}^{{2}}{+}{y}{+}{z}{-}{1}\right]\right]$ (3)

Solve the first and second equations.

 > $\mathrm{dec}≔\left[\mathrm{seq}\left(\mathrm{op}\left(\mathrm{Intersect}\left(\mathrm{sys}\left[2\right],\mathrm{rc},R\right)\right),\mathrm{rc}=\mathrm{dec}\right)\right];$$\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$
 $\left[\left[{x}{-}{y}{,}{{y}}^{{2}}{+}{y}{+}{z}{-}{1}\right]{,}\left[{x}{+}{y}{-}{1}{,}{{y}}^{{2}}{-}{y}{+}{z}\right]\right]$ (4)

Solve the three equations together.

 > $\mathrm{dec}≔\left[\mathrm{seq}\left(\mathrm{op}\left(\mathrm{Intersect}\left(\mathrm{sys}\left[3\right],\mathrm{rc},R\right)\right),\mathrm{rc}=\mathrm{dec}\right)\right];$$\mathrm{map}\left(\mathrm{Equations},\mathrm{dec},R\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}{,}{\mathrm{regular_chain}}\right]$
 $\left[\left[{x}{-}{z}{,}{y}{-}{z}{,}{{z}}^{{2}}{+}{2}{}{z}{-}{1}\right]{,}\left[{x}{,}{y}{,}{z}{-}{1}\right]{,}\left[{x}{-}{1}{,}{y}{,}{z}\right]{,}\left[{x}{,}{y}{-}{1}{,}{z}\right]\right]$ (5)

References

 Moreno Maza, M. "On Triangular Decompositions of Algebraic Varieties." MEGA-2000 conference. Bath, UK, England.