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.

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

$R≔\mathrm{differential\_ring}\left(\mathrm{derivations}=\left[x\,y\right]\,\mathrm{ranking}=\left[\mathrm{lex}\left[u,v\right]\right]\right)$

$R≔\mathrm{PDE\_ring}$

$p≔u\left[y\right]+v\left[\right]\:$

$q≔v\left[\right]{u\left[x\right]}^2+v\left[y,y\right]\:$

$\mathrm{leader}\left(p\,R\right),\mathrm{leader}\left(q\,R\right)$

$u_y,u_x$

$\mathrm{delta\_polynomial}\left(p\,q\,R\right)$

$u_x^2{v}_y-2u_{x}v\left[\right]v_x+v_{y,y,y}$