 IteratedResultantDim1 - Maple Help

RegularChains[FastArithmeticTools]

 IteratedResultantDim1
 iterated resultant of a polynomial w.r.t a one-dim regular chain Calling Sequence IteratedResultantDim1(f, rc, R, v) IteratedResultantDim1(f, rc, R, v, bound) Parameters

 R - a polynomial ring rc - a regular chain f - a polynomial v - variable of R bound - an upper bound of the degree of the iterated resultant to be computed (optional) Description

 • The function call IteratedResultantDim1(f, rc, R) returns the numerator of the iterated resultant of f w.r.t. rc, computed over the field of univariate rational functions in v and with coefficients in R. See the command IteratedResultant for a definition of the notion of an iterated resultant.
 • rc is assumed to be a one-dimensional normalized regular chain with v as free variable and f has positive degree w.r.t. v.
 • Moreover R must have a prime characteristic $p$ such that FFT-based polynomial arithmetic can be used for this actual computation. The higher the degrees of f and rc are, the larger must be $e$ such that ${2}^{e}$ divides $p-1$.  If the degree of  f or rc is too large, then an error is raised.
 • The default value of bound is the product of the total degrees of the polynomials in rc and f.
 • The iterated resultant computed by the command IteratedResultant produces the same answer provided that all initials in the regular chain rc are equal to $1$.
 • The interest of the function call IteratedResultantDim1(f, rc, R) resides in the fact that, if the polynomial f is regular modulo the saturated ideal of the regular chain rc, then the roots of the returned polynomial form the projection on the v-axis of the intersection of the hypersurface defined by f and the quasi-component defined by rc. Examples

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

Define a ring of polynomials.

 > $p≔469762049;$$\mathrm{vars}≔\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right];$$R≔\mathrm{PolynomialRing}\left(\mathrm{vars},p\right)$
 ${p}{≔}{469762049}$
 ${\mathrm{vars}}{≔}\left[{\mathrm{x1}}{,}{\mathrm{x2}}{,}{\mathrm{x3}}{,}{\mathrm{x4}}\right]$
 ${R}{≔}{\mathrm{polynomial_ring}}$ (1)

Define random dense polynomial and regular chain of R.

 > $N≔\mathrm{nops}\left(\mathrm{vars}\right):$$\mathrm{dg}≔3:$$\mathrm{degs}≔\left[\mathrm{seq}\left(2,i=1..N\right)\right]:$$\mathrm{pol}≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p;$$\mathrm{tc}≔\mathrm{RandomRegularChainDim1}\left(\mathrm{vars},\mathrm{degs},p\right);$$\mathrm{Equations}\left(\mathrm{tc},R\right)$
 ${\mathrm{pol}}{≔}{469762042}{}{{\mathrm{x1}}}^{{3}}{+}{22}{}{{\mathrm{x1}}}^{{2}}{}{\mathrm{x2}}{+}{469761994}{}{{\mathrm{x1}}}^{{2}}{}{\mathrm{x3}}{+}{469761955}{}{{\mathrm{x1}}}^{{2}}{}{\mathrm{x4}}{+}{469761993}{}{\mathrm{x1}}{}{{\mathrm{x2}}}^{{2}}{+}{469761987}{}{\mathrm{x1}}{}{\mathrm{x2}}{}{\mathrm{x4}}{+}{469761976}{}{\mathrm{x1}}{}{{\mathrm{x3}}}^{{2}}{+}{469762045}{}{\mathrm{x1}}{}{\mathrm{x3}}{}{\mathrm{x4}}{+}{469762039}{}{\mathrm{x1}}{}{{\mathrm{x4}}}^{{2}}{+}{80}{}{{\mathrm{x2}}}^{{3}}{+}{469762005}{}{{\mathrm{x2}}}^{{2}}{}{\mathrm{x3}}{+}{71}{}{{\mathrm{x2}}}^{{2}}{}{\mathrm{x4}}{+}{469761974}{}{\mathrm{x2}}{}{{\mathrm{x3}}}^{{2}}{+}{469762039}{}{\mathrm{x2}}{}{\mathrm{x3}}{}{\mathrm{x4}}{+}{469762009}{}{\mathrm{x2}}{}{{\mathrm{x4}}}^{{2}}{+}{23}{}{{\mathrm{x3}}}^{{3}}{+}{75}{}{{\mathrm{x3}}}^{{2}}{}{\mathrm{x4}}{+}{6}{}{\mathrm{x3}}{}{{\mathrm{x4}}}^{{2}}{+}{37}{}{{\mathrm{x4}}}^{{3}}{+}{87}{}{{\mathrm{x1}}}^{{2}}{+}{97}{}{\mathrm{x1}}{}{\mathrm{x2}}{+}{469761966}{}{\mathrm{x1}}{}{\mathrm{x3}}{+}{62}{}{\mathrm{x1}}{}{\mathrm{x4}}{+}{469762032}{}{{\mathrm{x2}}}^{{2}}{+}{469762042}{}{\mathrm{x2}}{}{\mathrm{x3}}{+}{42}{}{\mathrm{x2}}{}{\mathrm{x4}}{+}{469761957}{}{{\mathrm{x3}}}^{{2}}{+}{74}{}{\mathrm{x3}}{}{\mathrm{x4}}{+}{469762026}{}{{\mathrm{x4}}}^{{2}}{+}{469761967}{}{\mathrm{x1}}{+}{469761999}{}{\mathrm{x2}}{+}{72}{}{\mathrm{x3}}{+}{87}{}{\mathrm{x4}}{+}{23102807}$
 ${\mathrm{tc}}{≔}{\mathrm{regular_chain}}$
 $\left[{{\mathrm{x1}}}^{{2}}{+}{469761998}{}{\mathrm{x1}}{+}{77}{}{\mathrm{x2}}{+}{95}{}{\mathrm{x3}}{+}{\mathrm{x4}}{+}{377175716}{,}{{\mathrm{x2}}}^{{2}}{+}{40}{}{\mathrm{x2}}{+}{469761968}{}{\mathrm{x3}}{+}{91}{}{\mathrm{x4}}{+}{2502552}{,}{{\mathrm{x3}}}^{{2}}{+}{469762020}{}{\mathrm{x3}}{+}{95}{}{\mathrm{x4}}{+}{63792240}\right]$ (2)

Compute the (numerator) of the iterated resultant

 > $\mathrm{r1}≔\mathrm{IteratedResultantDim1}\left(\mathrm{pol},\mathrm{tc},R,\mathrm{x4}\right)$
 ${\mathrm{r1}}{≔}{68613548}{}{{\mathrm{x4}}}^{{24}}{+}{347134095}{}{{\mathrm{x4}}}^{{23}}{+}{360682950}{}{{\mathrm{x4}}}^{{22}}{+}{449975966}{}{{\mathrm{x4}}}^{{21}}{+}{452755530}{}{{\mathrm{x4}}}^{{20}}{+}{347383754}{}{{\mathrm{x4}}}^{{19}}{+}{223883343}{}{{\mathrm{x4}}}^{{18}}{+}{428024257}{}{{\mathrm{x4}}}^{{17}}{+}{190189697}{}{{\mathrm{x4}}}^{{16}}{+}{166005727}{}{{\mathrm{x4}}}^{{15}}{+}{88755441}{}{{\mathrm{x4}}}^{{14}}{+}{16726876}{}{{\mathrm{x4}}}^{{13}}{+}{30728041}{}{{\mathrm{x4}}}^{{12}}{+}{191794}{}{{\mathrm{x4}}}^{{11}}{+}{55677935}{}{{\mathrm{x4}}}^{{10}}{+}{232265645}{}{{\mathrm{x4}}}^{{9}}{+}{131365622}{}{{\mathrm{x4}}}^{{8}}{+}{100732316}{}{{\mathrm{x4}}}^{{7}}{+}{465359200}{}{{\mathrm{x4}}}^{{6}}{+}{463678220}{}{{\mathrm{x4}}}^{{5}}{+}{280061786}{}{{\mathrm{x4}}}^{{4}}{+}{453663429}{}{{\mathrm{x4}}}^{{3}}{+}{383524352}{}{{\mathrm{x4}}}^{{2}}{+}{254364287}{}{\mathrm{x4}}{+}{418973534}$ (3)

Compare with the generic algorithm (non-fast and non-modular algorithm) of the command IteratedResultant.

 > $\mathrm{r2}≔\mathrm{IteratedResultant}\left(\mathrm{pol},\mathrm{tc},R\right)$
 ${\mathrm{r2}}{≔}{68613548}{}{{\mathrm{x4}}}^{{24}}{+}{347134095}{}{{\mathrm{x4}}}^{{23}}{+}{360682950}{}{{\mathrm{x4}}}^{{22}}{+}{449975966}{}{{\mathrm{x4}}}^{{21}}{+}{452755530}{}{{\mathrm{x4}}}^{{20}}{+}{347383754}{}{{\mathrm{x4}}}^{{19}}{+}{223883343}{}{{\mathrm{x4}}}^{{18}}{+}{428024257}{}{{\mathrm{x4}}}^{{17}}{+}{190189697}{}{{\mathrm{x4}}}^{{16}}{+}{166005727}{}{{\mathrm{x4}}}^{{15}}{+}{88755441}{}{{\mathrm{x4}}}^{{14}}{+}{16726876}{}{{\mathrm{x4}}}^{{13}}{+}{30728041}{}{{\mathrm{x4}}}^{{12}}{+}{191794}{}{{\mathrm{x4}}}^{{11}}{+}{55677935}{}{{\mathrm{x4}}}^{{10}}{+}{232265645}{}{{\mathrm{x4}}}^{{9}}{+}{131365622}{}{{\mathrm{x4}}}^{{8}}{+}{100732316}{}{{\mathrm{x4}}}^{{7}}{+}{465359200}{}{{\mathrm{x4}}}^{{6}}{+}{463678220}{}{{\mathrm{x4}}}^{{5}}{+}{280061786}{}{{\mathrm{x4}}}^{{4}}{+}{453663429}{}{{\mathrm{x4}}}^{{3}}{+}{383524352}{}{{\mathrm{x4}}}^{{2}}{+}{254364287}{}{\mathrm{x4}}{+}{418973534}$ (4)

Check that the two results are equal, since here all initials are equal to 1.

 > $\mathrm{Expand}\left(\mathrm{r1}-\mathrm{r2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p$
 ${0}$ (5)