 RegularChains[ChainTools] - Maple Programming Help

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RegularChains[ChainTools]

 IteratedResultant
 return the iterated resultant of a polynomial with respect to a regular chain

 Calling Sequence IteratedResultant(p, rc, R)

Parameters

 p - polynomial of R rc - regular chain of R R - polynomial ring

Description

 • The command IteratedResultant(p, rc, R) returns the iterated resultant of p with respect to rc.
 • If p is a constant or all variables in p are free with respect to rc, then p is returned. Otherwise, if v is the largest variable of p algebraic with respect to rc, then IteratedResultant(r, Under(v, rc, R), R) is returned, where r is the resultant of p and the polynomial $\mathrm{Polynomial}\left(v,\mathrm{rc},R\right)$.
 • This command is part of the RegularChains[ChainTools] package, so it can be used in the form IteratedResultant(..) only after executing the command with(RegularChains[ChainTools]).  However, it can always be accessed through the long form of the command by using RegularChains[ChainTools][IteratedResultant](..).
 • The commands IteratedResultantDim0 and IteratedResultantDim1 provide fast algorithms for computing iterated resultants in prime characteristic and with regular chains in dimensions $0$ and $1$ respectively.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $R≔\mathrm{PolynomialRing}\left(\left[x,y,t,s\right]\right)$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)
 > $F≔\left[x+{y}^{2}-t,{t}^{2}-s\right]$
 ${F}{≔}\left[{{y}}^{{2}}{-}{t}{+}{x}{,}{{t}}^{{2}}{-}{s}\right]$ (2)
 > $p≔xyt$
 ${p}{≔}{x}{}{y}{}{t}$ (3)
 > $\mathrm{dec}≔\mathrm{Triangularize}\left(F,R\right)$
 ${\mathrm{dec}}{≔}\left[{\mathrm{regular_chain}}\right]$ (4)
 > $\mathrm{IteratedResultant}\left(p,\mathrm{dec}\left[1\right],R\right)$
 ${-}{s}{}{{y}}^{{6}}{+}{{s}}^{{2}}{}{{y}}^{{2}}$ (5)

 See Also