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RegularChains[FastArithmeticTools]

 RandomRegularChainDim1
 generate a random one-dim regular chain

 Calling Sequence RandomRegularChainDim1(lv, ld, p)

Parameters

 lv - a list of variables ld - a list of degrees p - a prime number

Description

 • The command RandomRegularChainDim1 returns a randomly generated one-dimensional regular chain with lv as variables. The degree sequence of the variables is ld. All the coefficients of the polynomials are reduced w.r.t p.

Examples

 > $\mathrm{with}\left(\mathrm{RegularChains}\right):$
 > $\mathrm{with}\left(\mathrm{ChainTools}\right):$
 > $\mathrm{with}\left(\mathrm{FastArithmeticTools}\right):$
 > $p≔962592769:$
 > $\mathrm{vars}≔\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right]:$
 > $R≔\mathrm{PolynomialRing}\left(\mathrm{vars},p\right):$

Randomly generating (dense) regular chain and polynomial

 > $N≔\mathrm{nops}\left(\mathrm{vars}\right):$
 > $\mathrm{dg}≔3:$
 > $\mathrm{degs}≔\left[\mathrm{seq}\left(2,i=1..N\right)\right]:$
 > $\mathrm{bound}≔{3}^{5}:$
 > $\mathrm{pol}≔\mathrm{randpoly}\left(\mathrm{vars},\mathrm{dense},\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}p\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}p:$
 > $\mathrm{tc}≔\mathrm{RandomRegularChainDim1}\left(\mathrm{vars},\mathrm{degs},p\right)$
 ${\mathrm{tc}}{:=}{\mathrm{regular_chain}}$ (1)

Computing with the modpn-supported and modular code

 > $\mathrm{r1}≔\mathrm{IteratedResultantDim1}\left(\mathrm{pol},\mathrm{tc},R,\mathrm{x4},\mathrm{bound}\right)$
 ${\mathrm{r1}}{:=}{941032609}{}{{\mathrm{x4}}}^{{24}}{+}{942118408}{}{{\mathrm{x4}}}^{{23}}{+}{194753142}{}{{\mathrm{x4}}}^{{22}}{+}{341712954}{}{{\mathrm{x4}}}^{{21}}{+}{2338033}{}{{\mathrm{x4}}}^{{20}}{+}{475384741}{}{{\mathrm{x4}}}^{{19}}{+}{457989148}{}{{\mathrm{x4}}}^{{18}}{+}{681536348}{}{{\mathrm{x4}}}^{{17}}{+}{718107114}{}{{\mathrm{x4}}}^{{16}}{+}{148212901}{}{{\mathrm{x4}}}^{{15}}{+}{33032177}{}{{\mathrm{x4}}}^{{14}}{+}{214227263}{}{{\mathrm{x4}}}^{{13}}{+}{150959749}{}{{\mathrm{x4}}}^{{12}}{+}{8508997}{}{{\mathrm{x4}}}^{{11}}{+}{76347793}{}{{\mathrm{x4}}}^{{10}}{+}{927296123}{}{{\mathrm{x4}}}^{{9}}{+}{733361705}{}{{\mathrm{x4}}}^{{8}}{+}{689644959}{}{{\mathrm{x4}}}^{{7}}{+}{204369026}{}{{\mathrm{x4}}}^{{6}}{+}{361192708}{}{{\mathrm{x4}}}^{{5}}{+}{138352162}{}{{\mathrm{x4}}}^{{4}}{+}{832291904}{}{{\mathrm{x4}}}^{{3}}{+}{263580044}{}{{\mathrm{x4}}}^{{2}}{+}{106333828}{}{\mathrm{x4}}{+}{323296554}$ (2)

Computing with the non-fast non-modular code

 > $\mathrm{r2}≔\mathrm{IteratedResultant}\left(\mathrm{pol},\mathrm{tc},R\right)$
 ${\mathrm{r2}}{:=}{941032609}{}{{\mathrm{x4}}}^{{24}}{+}{942118408}{}{{\mathrm{x4}}}^{{23}}{+}{194753142}{}{{\mathrm{x4}}}^{{22}}{+}{341712954}{}{{\mathrm{x4}}}^{{21}}{+}{2338033}{}{{\mathrm{x4}}}^{{20}}{+}{475384741}{}{{\mathrm{x4}}}^{{19}}{+}{457989148}{}{{\mathrm{x4}}}^{{18}}{+}{681536348}{}{{\mathrm{x4}}}^{{17}}{+}{718107114}{}{{\mathrm{x4}}}^{{16}}{+}{148212901}{}{{\mathrm{x4}}}^{{15}}{+}{33032177}{}{{\mathrm{x4}}}^{{14}}{+}{214227263}{}{{\mathrm{x4}}}^{{13}}{+}{150959749}{}{{\mathrm{x4}}}^{{12}}{+}{8508997}{}{{\mathrm{x4}}}^{{11}}{+}{76347793}{}{{\mathrm{x4}}}^{{10}}{+}{927296123}{}{{\mathrm{x4}}}^{{9}}{+}{733361705}{}{{\mathrm{x4}}}^{{8}}{+}{689644959}{}{{\mathrm{x4}}}^{{7}}{+}{204369026}{}{{\mathrm{x4}}}^{{6}}{+}{361192708}{}{{\mathrm{x4}}}^{{5}}{+}{138352162}{}{{\mathrm{x4}}}^{{4}}{+}{832291904}{}{{\mathrm{x4}}}^{{3}}{+}{263580044}{}{{\mathrm{x4}}}^{{2}}{+}{106333828}{}{\mathrm{x4}}{+}{323296554}$ (3)

The results computed by IteratedResultantDim1 and IteratedResultant are equivalent.

 > $\mathrm{evalb}\left(\mathrm{r1}=\mathrm{r2}\right)$
 ${\mathrm{true}}$ (4)