diffalg(deprecated)/preparation_polynomial - Help

diffalg

 preparation_polynomial
 compute preparation polynomial

 Calling Sequence preparation_polynomial (p, a, R, 'm' ) preparation_polynomial (p, A=a, R, 'm' )

Parameters

 p - differential polynomial in R a - regular differential polynomial in R R - differential polynomial ring m - (optional) name A - derivative of order zero in R

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • The function preparation_polynomial computes a preparation polynomial of p with respect to a.
 • The preparation polynomial of p with respect to a is a sort of expansion of p according to mparama and its derivatives. It plays a prominent role in the determination of the essential components of the radical differential ideal generated by a single differential polynomial.
 • A differential polynomial a is said to be regular if it has no common factor with its separant. This property is therefore dependent on the ranking defined on R.
 • If A is omitted, the preparation polynomial appears with an  indeterminate  (local variable) looking like  _A.
 • If A is  specified, the preparation polynomial is in the  differential indeterminate A. Then, A, nor its derivatives, should appear in p nor a.
 • Assume that preparation_polynomial(p, a, R, 'm') = $\mathrm{c1}\mathrm{M1}\left(\mathrm{_A}\right)+\mathrm{....}+\mathrm{ck}\mathrm{Mk}\left(\mathrm{_A}\right)$, where the Mi are differential monomials in $\mathrm{_A}$ and the $\mathrm{ci}$ are polynomials in R. Then
 - $mp=\mathrm{c1}\mathrm{M1}\left(a\right)+\mathrm{....}+\mathrm{ck}\mathrm{Mk}\left(a\right)$, where m belongs to R.
 - The $\mathrm{ci}$ are not reduced to zero by a, and therefore do not belong to the general component of a.
 - m is a power product of factors of the initial and separant of a).
 • The command with(diffalg,preparation_polynomial) allows the use of the abbreviated form of this command.

Examples

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

The preparation polynomial is used to determine the essential singular zeros of a differential polynomial.

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[u,A\right]\right):$
 > $p≔16{u}_{x,y}\left({u}_{x,x}^{2}-{u}_{y,y}^{2}+{u}_{y,y}-{u}_{x,x}\right)+{u}_{[]}^{2}\left(4{u}_{[]}-{x}^{2}-{y}^{2}\right)$
 ${p}{≔}{16}{}{{u}}_{{x}{,}{y}}{}\left({{u}}_{{x}{,}{x}}^{{2}}{-}{{u}}_{{y}{,}{y}}^{{2}}{-}{{u}}_{{x}{,}{x}}{+}{{u}}_{{y}{,}{y}}\right){+}{{u}}_{{[}{]}}^{{2}}{}\left({-}{{x}}^{{2}}{-}{{y}}^{{2}}{+}{4}{}{{u}}_{{[}{]}}\right)$ (1)
 > $\mathrm{equations}\left(\mathrm{Rosenfeld_Groebner}\left(\left[p\right],R\right)\right)$
 $\left[\left[{-}{{x}}^{{2}}{}{{u}}_{{[}{]}}^{{2}}{-}{{y}}^{{2}}{}{{u}}_{{[}{]}}^{{2}}{+}{4}{}{{u}}_{{[}{]}}^{{3}}{+}{16}{}{{u}}_{{x}{,}{x}}^{{2}}{}{{u}}_{{x}{,}{y}}{-}{16}{}{{u}}_{{x}{,}{y}}{}{{u}}_{{y}{,}{y}}^{{2}}{-}{16}{}{{u}}_{{x}{,}{x}}{}{{u}}_{{x}{,}{y}}{+}{16}{}{{u}}_{{x}{,}{y}}{}{{u}}_{{y}{,}{y}}\right]{,}\left[{-}{{x}}^{{2}}{-}{{y}}^{{2}}{+}{4}{}{{u}}_{{[}{]}}\right]{,}\left[{{u}}_{{[}{]}}\right]\right]$ (2)
 > $\mathrm{preparation_polynomial}\left(p,{u}_{[]},R\right)$
 ${4}{}{{\mathrm{_A}}}_{{[}{]}}^{{3}}{-}{16}{}{{\mathrm{_A}}}_{{y}{,}{y}}^{{2}}{}{{\mathrm{_A}}}_{{x}{,}{y}}{+}{16}{}{{\mathrm{_A}}}_{{x}{,}{y}}{}{{\mathrm{_A}}}_{{x}{,}{x}}^{{2}}{+}\left({-}{{x}}^{{2}}{-}{{y}}^{{2}}\right){}{{\mathrm{_A}}}_{{[}{]}}^{{2}}{+}{16}{}{{\mathrm{_A}}}_{{y}{,}{y}}{}{{\mathrm{_A}}}_{{x}{,}{y}}{-}{16}{}{{\mathrm{_A}}}_{{x}{,}{y}}{}{{\mathrm{_A}}}_{{x}{,}{x}}$ (3)
 > $\mathrm{preparation_polynomial}\left(p,{A}_{[]}=4{u}_{[]}-{x}^{2}-{y}^{2},R\right)$
 ${{A}}_{{[}{]}}^{{3}}{-}{4}{}{{A}}_{{y}{,}{y}}^{{2}}{}{{A}}_{{x}{,}{y}}{+}{4}{}{{A}}_{{x}{,}{y}}{}{{A}}_{{x}{,}{x}}^{{2}}{+}\left({2}{}{{x}}^{{2}}{+}{2}{}{{y}}^{{2}}\right){}{{A}}_{{[}{]}}^{{2}}{+}\left({{x}}^{{4}}{+}{2}{}{{x}}^{{2}}{}{{y}}^{{2}}{+}{{y}}^{{4}}\right){}{{A}}_{{[}{]}}$ (4)

Studying the degree in $A$ (or $\mathrm{_A}$) and its derivatives in these preparation polynomials, we can deduce that $u\left(x,y\right)=\frac{1}{4}{x}^{2}+\frac{1}{4}{y}^{2}$ is an essential singular zero of $p$ while $u\left(x,y\right)=0$ is not.

The preparation polynomial can be used to further study the relationships between the general zero and the singular zeros of $p$.

 > $R≔\mathrm{differential_ring}\left(\mathrm{ranking}=\left[y,A\right],\mathrm{derivations}=\left[x\right]\right):$
 > $p≔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$2}^{7}$
 ${p}{≔}{3}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}^{{2}}{-}{4}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{+}{6}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{12}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{3}}{+}{24}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{29}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{3}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{12}{}{{y}}_{{x}{,}{x}}^{{7}}$ (5)
 > $\mathrm{equations}\left(\mathrm{Rosenfeld_Groebner}\left(\left[p\right],R\right)\right)$
 $\left[\left[{3}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}^{{2}}{-}{4}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{+}{6}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{12}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{3}}{+}{24}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{29}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{3}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{12}{}{{y}}_{{x}{,}{x}}^{{7}}\right]{,}\left[{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{3}{}{{y}}_{{x}{,}{x}}^{{4}}\right]{,}\left[{{y}}_{{x}{,}{x}}\right]\right]$ (6)
 > $q≔3{y}_{x,x}^{4}+{y}_{x}^{2}{y}_{x,x,x}^{2}$
 ${q}{≔}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{3}{}{{y}}_{{x}{,}{x}}^{{4}}$ (7)
 > $\mathrm{preparation_polynomial}\left(p,{A}_{[]}=q,R\right)$
 $\left({-}{32}{}{{y}}_{{x}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}}{-}{32}{}{{y}}_{{x}{,}{x}}^{{3}}\right){}{{A}}_{{[}{]}}^{{2}}{-}{8}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}{}{{A}}_{{[}{]}}{}{{A}}_{{x}}{+}{3}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}}^{{2}}{}{{A}}_{{x}}^{{2}}{+}\left({96}{}{{y}}_{{x}}{}{{y}}_{{x}{,}{x}}^{{5}}{}{{y}}_{{x}{,}{x}{,}{x}}{+}{96}{}{{y}}_{{x}{,}{x}}^{{7}}\right){}{{A}}_{{[}{]}}$ (8)
 > $\mathrm{equations}\left(\mathrm{essential_components}\left(p,R\right)\right)$
 $\left[\left[{3}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}^{{2}}{-}{4}{}{{y}}_{{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{+}{6}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{12}{}{{y}}_{{x}}^{{3}}{}{{y}}_{{x}{,}{x}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{3}}{+}{24}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{4}}{}{{y}}_{{x}{,}{x}{,}{x}{,}{x}}{-}{29}{}{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}}^{{3}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{12}{}{{y}}_{{x}{,}{x}}^{{7}}\right]{,}\left[{{y}}_{{x}}^{{2}}{}{{y}}_{{x}{,}{x}{,}{x}}^{{2}}{+}{3}{}{{y}}_{{x}{,}{x}}^{{4}}\right]\right]$ (9)

The general zero of $q$ is an essential singular zero of $p$ while the general zero of ${y}_{x,x}$ is not. Thus, the straight lines $y\left(x\right)=\mathrm{_C1}x+\mathrm{_C2}$, zeros of ${y}_{x,x}$, must be limits of either some non singular zeros of $p$ or of the non singular zeros of $q$. Again studying the degrees of the preparation polynomials of $p$ and $q$ we can deduce that the straight lines are in fact limits of the non singular zeros of both (cf. [Kolchin]).

 > $\mathrm{preparation_polynomial}\left(p,{A}_{[]}={y}_{x,x},R\right)$
 ${12}{}{{A}}_{{[}{]}}^{{7}}{+}{24}{}{{A}}_{{[}{]}}^{{4}}{}{{A}}_{{x}{,}{x}}{}{{y}}_{{x}}^{{2}}{-}{29}{}{{A}}_{{[}{]}}^{{3}}{}{{A}}_{{x}}^{{2}}{}{{y}}_{{x}}^{{2}}{+}{6}{}{{A}}_{{[}{]}}^{{2}}{}{{A}}_{{x}}{}{{A}}_{{x}{,}{x}}{}{{y}}_{{x}}^{{3}}{-}{12}{}{{A}}_{{[}{]}}{}{{A}}_{{x}}^{{3}}{}{{y}}_{{x}}^{{3}}{+}{3}{}{{A}}_{{[}{]}}{}{{A}}_{{x}{,}{x}}^{{2}}{}{{y}}_{{x}}^{{4}}{-}{4}{}{{A}}_{{x}}^{{2}}{}{{A}}_{{x}{,}{x}}{}{{y}}_{{x}}^{{4}}$ (10)
 > $\mathrm{preparation_polynomial}\left(q,{A}_{[]}={y}_{x,x},R\right)$
 ${3}{}{{A}}_{{[}{]}}^{{4}}{+}{{A}}_{{x}}^{{2}}{}{{y}}_{{x}}^{{2}}$ (11)