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 .

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

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

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

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

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

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

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{dec}≔\mathrm{Intersect}\left(\mathrm{sys}\left[1\right]\,\mathrm{rc}\,R\right)\;$$\mathrm{map}\left(\mathrm{Equations}\,\mathrm{dec}\,R\right)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\right]$

$\left[\left[x^2+y+z-1\right]\right]$

$\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)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\left[\left[x-y\,y^2+y+z-1\right]\,\left[x+y-1\,y^2-y+z\right]\right]$

$\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)$

$\mathrm{dec}≔\left[\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\,\mathrm{regular\_chain}\right]$

$\left[\left[x-z\,y-z\,z^2+2z-1\right]\,\left[x\,y\,z-1\right]\,\left[x-1\,y\,z\right]\,\left[x\,y-1\,z\right]\right]$

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