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DifferentialAlgebra[Tools]

 DeltaPolynomial
 returns a Delta-polynomial

 Calling Sequence DeltaPolynomial (p, q, R,opts)

Parameters

 p - a differential polynomial q - a differential polynomial R - a differential polynomial ring or ideal opts (optional) - a sequence of options

Options

 • The opts arguments may contain one or more of the options below.
 • notation = jet, tjet, diff or Diff. Specifies the notation used for the result of the function call. If not specified, the notation of R or of ideal is used.
 • memout = nonnegative. Specifies a memory limit, in MB, for the computation. Default is zero (no memory out).

Description

 • The function call DeltaPolynomial (p, q, R) returns the $\mathrm{\Delta }$-polynomial generated by p and q, which are regarded as differential polynomials of R, or, of its embedding ring, if R is an ideal. See DifferentialAlgebra for the definition of $\mathrm{\Delta }$-polynomials.
 • The numeric coefficients of the returned $\mathrm{\Delta }$-polynomial are normalized: their gcd is equal to $1$, and, the leading one is positive. It is required that the leading derivatives of p and q are derivatives of some same dependent variable.
 • This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form DeltaPolynomial(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][DeltaPolynomial](...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):$$\mathrm{with}\left(\mathrm{Tools}\right):$
 > $R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=\left[x,y\right],\mathrm{blocks}=\left[u,v\right]\right)$
 ${R}{≔}{\mathrm{differential_ring}}$ (1)

The triangular case: the least common derivative of the two leading derivatives is different from both of them.

 > $\mathrm{DeltaPolynomial}\left(u\left[x\right]-v,u\left[y\right],R\right)$
 ${{v}}_{{y}}$ (2)

The non-triangular case: the leading derivative of the second argument is a derivative of the leading derivative of the first one.

 > $\mathrm{DeltaPolynomial}\left({u\left[x\right]}^{2}-4u,u\left[x,x\right],R\right)$
 ${{u}}_{{x}}$ (3)