diffalg(deprecated)/essential_components - Help

diffalg

 essential_components
 compute a minimal characteristic decomposition

 Calling Sequence essential_components (p, R)

Parameters

 p - differential polynomial in R R - differential polynomial ring

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • The function essential_components returns a minimal characteristic decomposition of the radical  differential ideal generated by the single differential polynomial p.
 • Each of the characterizable components returned has a characteristic set consisting of only one differential polynomial, say $\mathrm{a1},\mathrm{...},\mathrm{ak}$.
 This means that the set of solutions of the differential equation $p=0$ is minimally described as the union of the general solutions of $\mathrm{a1}=0$, ... , $\mathrm{ak}=0$.
 The set of irreducible factors of $\mathrm{a1},\mathrm{...},\mathrm{ak}$ does not depend on the ranking chosen for R.
 • This function proceeds by eliminating the redundancy in the characteristic decomposition computed by Rosenfeld_Groebner applied to ([p], R).
 • The command with(diffalg,essential_components) allows the use of the abbreviated form of this command.

Examples

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$

Ordinary differential polynomials of first order:

 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[t\right],\mathrm{ranking}=\left[y\right],\mathrm{notation}=\mathrm{diff}\right):$
 > $p≔{\left(\frac{ⅆ}{ⅆt}y\left(t\right)\right)}^{3}-4ty\left(t\right)\left(\frac{ⅆ}{ⅆt}y\left(t\right)\right)+8{y\left(t\right)}^{2}$
 ${p}{≔}{\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right)}^{{3}}{-}{4}{}{t}{}{y}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right){+}{8}{}{{y}{}\left({t}\right)}^{{2}}$ (1)
 > $\mathrm{equations}\left(\mathrm{Rosenfeld_Groebner}\left(\left[p\right],R\right)\right)$
 $\left[\left[{\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right)}^{{3}}{-}{4}{}{t}{}{y}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right){+}{8}{}{{y}{}\left({t}\right)}^{{2}}\right]{,}\left[{27}{}{y}{}\left({t}\right){-}{4}{}{{t}}^{{3}}\right]{,}\left[{y}{}\left({t}\right)\right]\right]$ (2)

This differential polynomial has two singular zeros: the cubic $y\left(t\right)=\frac{4}{27}{t}^{3}$ and $y\left(t\right)=0$. Nonetheless, the general zero can be expressed as $y\left(t\right)=\mathrm{_C}{\left(t-\mathrm{_C}\right)}^{2}$. Therefore, $y\left(t\right)=0$ is a particular case ($\mathrm{_C}=0$) of the general solution. This is uncovered by essential_components without solving the differential equation. The function essential_components gives a minimal description of the zero set.

 > $\mathrm{equations}\left(\mathrm{essential_components}\left(p,R\right)\right)$
 $\left[\left[{\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right)}^{{3}}{-}{4}{}{t}{}{y}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right){+}{8}{}{{y}{}\left({t}\right)}^{{2}}\right]{,}\left[{27}{}{y}{}\left({t}\right){-}{4}{}{{t}}^{{3}}\right]\right]$ (3)

Let us consider the two similar differential polynomials $p$ and $q$.

 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[t\right],\mathrm{ranking}=\left[y\right]\right):$
 > $p≔{y}_{t}^{2}-4{y}_{[]}$
 ${p}{≔}{{y}}_{{t}}^{{2}}{-}{4}{}{{y}}_{{[}{]}}$ (4)
 > $q≔{y}_{t}^{2}-4{y}_{[]}^{3}$
 ${q}{≔}{-}{4}{}{{y}}_{{[}{]}}^{{3}}{+}{{y}}_{{t}}^{{2}}$ (5)
 > $\mathrm{Cp}≔\mathrm{equations}\left(\mathrm{Rosenfeld_Groebner}\left(\left[p\right],R\right)\right)$
 ${\mathrm{Cp}}{≔}\left[\left[{{y}}_{{t}}^{{2}}{-}{4}{}{{y}}_{{[}{]}}\right]{,}\left[{{y}}_{{[}{]}}\right]\right]$ (6)
 > $\mathrm{Cq}≔\mathrm{equations}\left(\mathrm{Rosenfeld_Groebner}\left(\left[q\right],R\right)\right)$
 ${\mathrm{Cq}}{≔}\left[\left[{-}{4}{}{{y}}_{{[}{]}}^{{3}}{+}{{y}}_{{t}}^{{2}}\right]{,}\left[{{y}}_{{[}{]}}\right]\right]$ (7)

Both $p$ and $q$ admit $y\left(t\right)=0$ as a singular zero. Nonetheless:

 > $\mathrm{Mp}≔\mathrm{equations}\left(\mathrm{essential_components}\left(p,R\right)\right)$
 ${\mathrm{Mp}}{≔}\left[\left[{{y}}_{{t}}^{{2}}{-}{4}{}{{y}}_{{[}{]}}\right]{,}\left[{{y}}_{{[}{]}}\right]\right]$ (8)
 > $\mathrm{Mq}≔\mathrm{equations}\left(\mathrm{essential_components}\left(q,R\right)\right)$
 ${\mathrm{Mq}}{≔}\left[\left[{-}{4}{}{{y}}_{{[}{]}}^{{3}}{+}{{y}}_{{t}}^{{2}}\right]\right]$ (9)

$y\left(t\right)=0$ is an essential singular zero of $p$ but not of $q$. This has an analytic interpretation: $y\left(t\right)=0$ is an envelope of the non singular zeros of $p$ while it is a limit of the non singular zeros of $q$.

Incidentally: the general zero of $q$ can be expressed as $y\left(t\right)=\frac{\mathrm{_C}}{{\left(\mathrm{_C}t-1\right)}^{2}}$. Thus, $y\left(t\right)=0$ is a particular case of the general zero of $q$.

Partial differential polynomials:

This illustrates the fact that the characteristic sets of the components of the minimal characteristic decomposition have only one element.

 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[u\right]\right):$
 > $p≔-{u}_{[]}+y{u}_{y}+x{u}_{x}-{u}_{x}^{2}-{u}_{y}^{2}$
 ${p}{≔}{x}{}{{u}}_{{x}}{+}{y}{}{{u}}_{{y}}{-}{{u}}_{{x}}^{{2}}{-}{{u}}_{{y}}^{{2}}{-}{{u}}_{{[}{]}}$ (10)
 > $C≔\mathrm{equations}\left(\mathrm{Rosenfeld_Groebner}\left(\left[p\right],R\right)\right)$
 ${C}{≔}\left[\left[{-}{x}{}{{u}}_{{x}}{-}{y}{}{{u}}_{{y}}{+}{{u}}_{{x}}^{{2}}{+}{{u}}_{{y}}^{{2}}{+}{{u}}_{{[}{]}}\right]{,}\left[{2}{}{{u}}_{{x}}{-}{x}{,}{-}{{x}}^{{2}}{-}{4}{}{y}{}{{u}}_{{y}}{+}{4}{}{{u}}_{{y}}^{{2}}{+}{4}{}{{u}}_{{[}{]}}\right]{,}\left[{-}{{x}}^{{2}}{-}{{y}}^{{2}}{+}{4}{}{{u}}_{{[}{]}}\right]\right]$ (11)
 > $M≔\mathrm{equations}\left(\mathrm{essential_components}\left(p,R\right)\right)$
 ${M}{≔}\left[\left[{-}{x}{}{{u}}_{{x}}{-}{y}{}{{u}}_{{y}}{+}{{u}}_{{x}}^{{2}}{+}{{u}}_{{y}}^{{2}}{+}{{u}}_{{[}{]}}\right]{,}\left[{-}{{x}}^{{2}}{-}{{y}}^{{2}}{+}{4}{}{{u}}_{{[}{]}}\right]\right]$ (12)

A differential polynomial in several variables:

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[u,v\right]\right):$
 > $p≔{u}_{x,y}^{2}{v}_{y}-{u}_{x,y}{v}_{y}{u}_{y}-{u}_{y}{u}_{x,y}+{u}_{y}^{2}$
 ${p}{≔}{-}{{u}}_{{y}}{}{{u}}_{{x}{,}{y}}{}{{v}}_{{y}}{+}{{u}}_{{x}{,}{y}}^{{2}}{}{{v}}_{{y}}{+}{{u}}_{{y}}^{{2}}{-}{{u}}_{{y}}{}{{u}}_{{x}{,}{y}}$ (13)
 > $\mathrm{Rosenfeld_Groebner}\left(\left[p\right],R\right)$
 $\left[{\mathrm{characterizable}}{,}{\mathrm{characterizable}}\right]$ (14)

It would seem that there several types of zeros, the general zero of $p$ and several singular zeros. Nonetheless,

 > $\mathrm{MR}≔\mathrm{essential_components}\left(p,R\right)$
 ${\mathrm{MR}}{≔}\left[{\mathrm{characterizable}}\right]$ (15)
 > $\mathrm{ER}≔\mathrm{equations}\left(\mathrm{MR}\right)$
 ${\mathrm{ER}}{≔}\left[\left[{-}{{u}}_{{y}}{}{{u}}_{{x}{,}{y}}{}{{v}}_{{y}}{+}{{u}}_{{x}{,}{y}}^{{2}}{}{{v}}_{{y}}{+}{{u}}_{{y}}^{{2}}{-}{{u}}_{{y}}{}{{u}}_{{x}{,}{y}}\right]\right]$ (16)

This show that the singular zeros exhibited by the Rosenfeld_Groebner decomposition are in fact particular zeros of the general zero of $p$.

We illustrate now the fact that the underlying prime minimal decomposition of the obtained characteristic minimal decomposition is independent of the ranking.

 > $Q≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[v,u\right]\right):$
 > $\mathrm{MQ}≔\mathrm{essential_components}\left(p,Q\right)$
 ${\mathrm{MQ}}{≔}\left[{\mathrm{characterizable}}\right]$ (17)
 > $\mathrm{EQ}≔\mathrm{equations}\left(\mathrm{MQ}\right)$
 ${\mathrm{EQ}}{≔}\left[\left[{{u}}_{{x}{,}{y}}{}{{v}}_{{y}}{-}{{u}}_{{y}}\right]\right]$ (18)

We check that the two differential polynomials appearing in this decompositions are the two factors of differential polynomials appearing in $\mathrm{MR}$.

 > $\mathrm{factor}\left({{\mathrm{ER}}_{1}}_{1}\right)$
 $\left({{u}}_{{y}}{-}{{u}}_{{x}{,}{y}}\right){}\left({-}{{u}}_{{x}{,}{y}}{}{{v}}_{{y}}{+}{{u}}_{{y}}\right)$ (19)

Higher order differential polynomials:

The following equation arose in Chazy's work to extend the Painleve analysis to third order differential equations. In the process, he uncovered certain differential equations whose non-singular solutions have no movable singularity whereas one of the singular solutions does.

 > $R≔\mathrm{differential_ring}\left(\mathrm{ranking}=\left[y\right],\mathrm{derivations}=\left[x\right]\right):$
 > $\mathrm{chazy}≔-{\left({y}_{x,x}+{y}_{[]}^{3}{y}_{x}\right)}^{2}+{\left({y}_{[]}{y}_{x}\right)}^{2}\left(4{y}_{x}+{y}_{[]}^{4}\right)$
 ${\mathrm{chazy}}{≔}{-}{\left({{y}}_{{[}{]}}^{{3}}{}{{y}}_{{x}}{+}{{y}}_{{x}{,}{x}}\right)}^{{2}}{+}{{y}}_{{[}{]}}^{{2}}{}{{y}}_{{x}}^{{2}}{}\left({{y}}_{{[}{]}}^{{4}}{+}{4}{}{{y}}_{{x}}\right)$ (20)
 > $\mathrm{equations}\left(\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{chazy}\right],R\right)\right)$
 $\left[\left[{2}{}{{y}}_{{[}{]}}^{{3}}{}{{y}}_{{x}}{}{{y}}_{{x}{,}{x}}{-}{4}{}{{y}}_{{[}{]}}^{{2}}{}{{y}}_{{x}}^{{3}}{+}{{y}}_{{x}{,}{x}}^{{2}}\right]{,}\left[{{y}}_{{[}{]}}^{{4}}{+}{4}{}{{y}}_{{x}}\right]{,}\left[{{y}}_{{x}}\right]\right]$ (21)
 > $\mathrm{equations}\left(\mathrm{essential_components}\left(\mathrm{chazy},R\right)\right)$
 $\left[\left[{2}{}{{y}}_{{[}{]}}^{{3}}{}{{y}}_{{x}}{}{{y}}_{{x}{,}{x}}{-}{4}{}{{y}}_{{[}{]}}^{{2}}{}{{y}}_{{x}}^{{3}}{+}{{y}}_{{x}{,}{x}}^{{2}}\right]{,}\left[{{y}}_{{[}{]}}^{{4}}{+}{4}{}{{y}}_{{x}}\right]\right]$ (22)

The singular zeros are given by $y\left(x\right)=\mathrm{_C}$ and ${y\left(x\right)}^{3}=\frac{1}{\frac{3}{4}x+\mathrm{_C}}$. Only the second kind is essential.

The zeros of the following 4th order, homogeneous differential equation of degree 7 have the property that they can be used to approximate piecewisely any smooth function. This was shown  by Rubel (1981).

 > $R≔\mathrm{differential_ring}\left(\mathrm{ranking}=\left[y,z\right],\mathrm{derivations}=\left[x\right]\right):$
 > $\mathrm{rubel}≔3{y}_{x}^{4}{y}_{x$2}{y}_{x$4}^{2}-4{y}_{x}^{4}{y}_{x$3}^{2}{y}_{x$4}+6{y}_{x}^{3}{y}_{x$2}^{2}{y}_{x$3}{y}_{x$4}+24{y}_{x}^{2}{y}_{x$2}^{4}{y}_{x$4}-12{y}_{x}^{3}{y}_{x$2}{y}_{x$3}^{3}-29{y}_{x}^{2}{y}_{x$2}^{3}{y}_{x$3}^{2}+12{y}_{x$2}^{7}$
 ${\mathrm{rubel}}{≔}{3}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}^{{2}}{-}{4}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{+}{6}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{12}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{3}}{+}{24}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{29}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{3}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{12}{}{{y}}_{{x}{,}{x}}^{{7}}$ (23)
 > $\mathrm{equations}\left(\mathrm{Rosenfeld_Groebner}\left(\left[\mathrm{rubel}\right],R\right)\right)$
 $\left[\left[{3}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}^{{2}}{-}{4}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{+}{6}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{12}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{3}}{+}{24}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{29}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{3}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{12}{}{{y}}_{{x}{,}{x}}^{{7}}\right]{,}\left[{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{3}{}{{y}}_{{x}{,}{x}}^{{4}}\right]{,}\left[{{y}}_{{x}{,}{x}}\right]\right]$ (24)
 > $\mathrm{equations}\left(\mathrm{essential_components}\left(\mathrm{rubel},R\right)\right)$
 $\left[\left[{3}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}^{{2}}{-}{4}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{+}{6}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{12}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{3}}{+}{24}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{29}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{3}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{12}{}{{y}}_{{x}{,}{x}}^{{7}}\right]{,}\left[{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{3}{}{{y}}_{{x}{,}{x}}^{{4}}\right]\right]$ (25)