The command RegularChainBranches(rc, R, L) returns the Puiseux series solutions of the regular chain rc around each of the roots of the polynomials in L.

The returned Puiseux series are actually truncated at a precision which is high enough to separate the different branches associated with the same root of a polynomial in L.

It is assumed that the coefficient field of R is the field of rational numbers.

It is assumed that rc is a one-dimensional strongly normalized regular chain.

This implies that every initial of a polynomial f in rc is either constant or univariate in a variable, say v, of R which is not algebraic w.r.t. rc.

It is assumed that the polynomials in L are univariate in that variable v. Moreover, it is assumed that every root of a polynomial in L is a root of the initial of a polynomial in rc.

Finally, it is assumed that V is an unassigned variable name.

The output of the command RegularChainBranches(rc, R, L) is a list of tuples. Each tuple is a Puiseux series solution of rc (associated with a root of the initial of a polynomial in rc) using an auxiliary variable to represent each Puiseux series as a pair of two polynomials; the first polynomial in such a pair is of a power of a variable (which is V if it is specified, otherwise T) and the second polynomial is univariate in that same variable.

In each Puiseux series solution of rc, each variable of R is given as a Puiseux series and all those Puiseux series share the same first polynomial in their representation; therefore, this polynomial appears only once as the first item in the tuple encoding this Puiseux series solution of rc.

If the option coefficient=real is present, then only the Puiseux series solutions with real coefficients are returned. Note those real Puiseux solutions are independent of the precision which is used in the returned truncated Puiseux series. In other words, the underlying algorithm guarantees that the returned Puiseux solutions represent all the real branches.

The accuracies option is for advanced users. If the option accuracies = L is present, where L is a list of integers, then L needs to have the same number of entries as the number of polynomials in rc. The effect is that the number of terms used for the ith polynomial in rc is L[i]. Any value used must comply with a certain set of requirements set out in the paper by Alvandi, Chen, and Moreno Maza listed in the References section.

If the accuracies option is not present, then a default value is computed. This value complies with the requirements listed in the paper.

This command is part of the RegularChains[AlgebraicGeometryTools] package, so it can be used in the form RegularChainBranches(..) only after executing the command with(RegularChains[AlgebraicGeometryTools]). However, it can always be accessed through the long form of the command by using RegularChains[AlgebraicGeometryTools][RegularChainBranches](..).

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

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

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[-z^2+y\,xz-y^2\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Display}\left(\mathrm{rc}\,R\right)$

$\left\{\begin{array}{cc}zx-y^2=0& \\ y-z^2=0& \\ z\ne 0& \end{array}\right.$

$\mathrm{br}≔\mathrm{RegularChainBranches}\left(\mathrm{rc}\,R\,\left[z\right]\right)$

$\mathrm{br}≔\left[\left[z=\mathrm{\_T}\,y={\mathrm{\_T}}^2\,x={\mathrm{\_T}}^3\right]\right]$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[y^{2}z+y+1\,\left(z+2\right)z{x}^2+\left(y+1\right)\left(x+1\right)\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Display}\left(\mathrm{rc}\,R\right)$

$\left\{\begin{array}{cc}\left(z^2+2z\right)x^2+\left(y+1\right)x+y+1=0& \\ z{y}^2+y+1=0& \\ z\ne 0& \\ z^2+2z\ne 0& \end{array}\right.$

$\mathrm{RegularChainBranches}\left(\mathrm{rc}\,R\,\left[z\right]\right)$

$\left[\left[z=\mathrm{\_T0}\,y=-\mathrm{\_T0}-1\,x=-\frac{1}{432}{\mathrm{\_T0}}^5+\frac{11}{432}{\mathrm{\_T0}}^4+\frac{5}{432}{\mathrm{\_T0}}^3-\frac{5}{216}{\mathrm{\_T0}}^2+\frac{1}{12}\mathrm{\_T0}-\frac{1}{2}\right]\,\left[z=\mathrm{\_T0}\,y=-\mathrm{\_T0}-1\,x=\frac{\left(\mathrm{\_T0}-2\right)\left({\mathrm{\_T0}}^2+4\right)\left({\mathrm{\_T0}}^2-9\mathrm{\_T0}-54\right)}{432}\right]\right]$

$\mathrm{RegularChainBranches}\left(\mathrm{rc}\,R\,\left[z+2\right]\right)$

$\left[\left[z=\mathrm{\_T1}-2\,y=-\frac{1}{216}{\mathrm{\_T1}}^4-\frac{11}{216}{\mathrm{\_T1}}^3+\frac{4}{27}{\mathrm{\_T1}}^2+\frac{1}{3}\mathrm{\_T1}+1\,x=\frac{3}{8}{\mathrm{\_T1}}^3+\frac{1}{2}{\mathrm{\_T1}}^2+\mathrm{\_T1}-1\right]\,\left[z=\mathrm{\_T1}-2\,y=\frac{1}{216}{\mathrm{\_T1}}^4+\frac{11}{216}{\mathrm{\_T1}}^3-\frac{5}{216}{\mathrm{\_T1}}^2-\frac{1}{12}\mathrm{\_T1}-\frac{1}{2}\,x=\frac{9}{8}{\mathrm{\_T1}}^3+2{\mathrm{\_T1}}^2+4\mathrm{\_T1}-1\right]\right]$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[y^3-2y^3+y^2+z^5\,z^{4}x+y^3-y^2\right]\,\mathrm{Empty}\left(R\right)\,R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{Display}\left(\mathrm{rc}\,R\right)$

$\left\{\begin{array}{cc}{z}^{4}x+y^3-y^2=0& \\ -y^3+y^2+z^5=0& \\ z^4\ne 0& \end{array}\right.$

$\mathrm{br}≔\mathrm{RegularChainBranches}\left(\mathrm{rc}\,R\,\left[z\right]\right)$

$\mathrm{br}≔\left[\left[z={\mathrm{\_T2}}^2\,y=-\frac{{\mathrm{\_T2}}^5\left({\mathrm{\_T2}}^5+2\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)}{2}\,x=\frac{{\mathrm{\_T2}}^2\left({\mathrm{\_T2}}^{20}+6{\mathrm{\_T2}}^{15}\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)-10{\mathrm{\_T2}}^{10}-8\right)}{8}\right]\,\left[z={\mathrm{\_T2}}^2\,y=-\frac{{\mathrm{\_T2}}^5\left({\mathrm{\_T2}}^5-2\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)\right)}{2}\,x=\frac{{\mathrm{\_T2}}^2\left({\mathrm{\_T2}}^{20}-6{\mathrm{\_T2}}^{15}\mathrm{RootOf}\left({\mathrm{\_Z}}^2+1\right)-10{\mathrm{\_T2}}^{10}-8\right)}{8}\right]\,\left[z=\mathrm{\_T2}\,y={\mathrm{\_T2}}^5+1\,x=-{\mathrm{\_T2}}^{11}-2{\mathrm{\_T2}}^6-\mathrm{\_T2}\right]\right]$

$\mathrm{br}≔\mathrm{RegularChainBranches}\left(\mathrm{rc}\,R\,\left[z\right]\,\mathrm{coefficient}=\mathrm{real}\right)$

$\mathrm{br}≔\left[\left[z=\mathrm{\_T3}\,y={\mathrm{\_T3}}^5+1\,x=-{\mathrm{\_T3}}^{11}-2{\mathrm{\_T3}}^6-\mathrm{\_T3}\right]\right]$

Parisa Alvandi, Changbo Chen, Marc Moreno Maza "Computing the Limit Points of the Quasi-component of a Regular Chain in Dimension One." Computer Algebra in Scientific Computing (CASC), Lecture Notes in Computer Science - 8136, (2013): 30-45.

Parisa Alvandi, Masoud Ataei, Mahsa Kazemi, Marc Moreno Maza "On the Extended Hensel Construction and its application to the computation of real limit points." J. Symb. Comput. 98: 120-162 (2020)

The RegularChains[AlgebraicGeometryTools][RegularChainBranches] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.