diffalg(deprecated)/delta_polynomial - Maple Help
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diffalg

 delta_polynomial
 return the delta-polynomial generated by two differential polynomials

 Calling Sequence delta_polynomial (p, q, R)

Parameters

 p, q - differential polynomials in R R - differential polynomial ring

Description

 • Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
 • The delta_polynomial command returns the delta-polynomial of p and q, that is
 $\mathrm{separant}\left(q,R\right)\mathrm{\phi }\left(p\right)-\mathrm{separant}\left(p,R\right)\mathrm{\psi }\left(q\right)$
 where phi and psi  are  the derivation operators of least order such that phi (up) = psi (uq) where up and uq are the leaders of p and q.
 The delta-polynomial is sometimes called the cross-derivative.
 • The differential polynomials p and q must not belong to the ground field of R. Their leaders must be derivatives of the same differential indeterminate but not be derivatives of each other. Otherwise, the delta-polynomial is not defined and delta_polynomial returns an error message.
 • Delta-polynomials are constructed when dealing with partial differential polynomials to obtain regular differential systems.
 • The command with(diffalg,delta_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.

 > $\mathrm{with}\left(\mathrm{diffalg}\right):$
 > $R≔\mathrm{differential_ring}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{ranking}=\left[\mathrm{lex}\left[u,v\right]\right]\right)$
 ${R}{≔}{\mathrm{PDE_ring}}$ (1)
 > $p≔u\left[y\right]+v\left[\right]:$
 > $q≔v\left[\right]{u\left[x\right]}^{2}+v\left[y,y\right]:$
 > $\mathrm{leader}\left(p,R\right),\mathrm{leader}\left(q,R\right)$
 ${{u}}_{{y}}{,}{{u}}_{{x}}$ (2)
 > $\mathrm{delta_polynomial}\left(p,q,R\right)$
 ${{u}}_{{x}}^{{2}}{}{{v}}_{{y}}{-}{2}{}{{u}}_{{x}}{}{v}\left[\right]{}{{v}}_{{x}}{+}{{v}}_{{y}{,}{y}{,}{y}}$ (3)