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.

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

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

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

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

$p≔469762049$

$\mathrm{vars}≔\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]$

$R≔\mathrm{polynomial\_ring}$

$N≔\mathrm{nops}\left(\mathrm{vars}\right)\:$$\mathrm{dg}≔3\:$$\mathrm{degs}≔\left[\mathrm{seq}\left(2\,i=1..N\right)\right]\:$$\mathrm{pol}≔\mathrm{randpoly}\left(\mathrm{vars}\,\mathrm{dense}\,\mathrm{degree}=\mathrm{dg}\right)+\left(\mathrm{rand}\left(\right)\mathbf{mod}p\right)\mathbf{mod}p\;$$\mathrm{tc}≔\mathrm{RandomRegularChainDim1}\left(\mathrm{vars}\,\mathrm{degs}\,p\right)\;$$\mathrm{Equations}\left(\mathrm{tc}\,R\right)$

$\mathrm{pol}≔469762042{\mathrm{x1}}^3+22{\mathrm{x1}}^2\mathrm{x2}+469761994{\mathrm{x1}}^2\mathrm{x3}+469761955{\mathrm{x1}}^2\mathrm{x4}+469761993\mathrm{x1}{\mathrm{x2}}^2+469761987\mathrm{x1}\mathrm{x2}\mathrm{x4}+469761976\mathrm{x1}{\mathrm{x3}}^2+469762045\mathrm{x1}\mathrm{x3}\mathrm{x4}+469762039\mathrm{x1}{\mathrm{x4}}^2+80{\mathrm{x2}}^3+469762005{\mathrm{x2}}^2\mathrm{x3}+71{\mathrm{x2}}^2\mathrm{x4}+469761974\mathrm{x2}{\mathrm{x3}}^2+469762039\mathrm{x2}\mathrm{x3}\mathrm{x4}+469762009\mathrm{x2}{\mathrm{x4}}^2+23{\mathrm{x3}}^3+75{\mathrm{x3}}^2\mathrm{x4}+6\mathrm{x3}{\mathrm{x4}}^2+37{\mathrm{x4}}^3+87{\mathrm{x1}}^2+97\mathrm{x1}\mathrm{x2}+469761966\mathrm{x1}\mathrm{x3}+62\mathrm{x1}\mathrm{x4}+469762032{\mathrm{x2}}^2+469762042\mathrm{x2}\mathrm{x3}+42\mathrm{x2}\mathrm{x4}+469761957{\mathrm{x3}}^2+74\mathrm{x3}\mathrm{x4}+469762026{\mathrm{x4}}^2+469761967\mathrm{x1}+469761999\mathrm{x2}+72\mathrm{x3}+87\mathrm{x4}+23102807$

$\mathrm{tc}≔\mathrm{regular\_chain}$

$\left[{\mathrm{x1}}^2+469761998\mathrm{x1}+77\mathrm{x2}+95\mathrm{x3}+\mathrm{x4}+377175716\,{\mathrm{x2}}^2+40\mathrm{x2}+469761968\mathrm{x3}+91\mathrm{x4}+2502552\,{\mathrm{x3}}^2+469762020\mathrm{x3}+95\mathrm{x4}+63792240\right]$

$\mathrm{r1}≔\mathrm{IteratedResultantDim1}\left(\mathrm{pol}\,\mathrm{tc}\,R\,\mathrm{x4}\right)$

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

$\mathrm{r2}≔\mathrm{IteratedResultant}\left(\mathrm{pol}\,\mathrm{tc}\,R\right)$

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

$\mathrm{Expand}\left(\mathrm{r1}-\mathrm{r2}\right)\mathbf{mod}p$

$0$