The function PolynomialToPDE maps polynomials to linear homogeneous PDEs and has as its inverse PDEToPolynomial. These maps  implement an equivalence between polynomial ideals (and in general modules), and systems of linear homogeneous constant coefficient PDEs.

These functions widen the set of tools that can be applied to both domains. For example, Groebner-like differential elimination algorithms, correspond to Groebner bases of modules under PolynomialToPDE. Efficiency features such as case-splitting in the differential packages, restricting time limits on the cases,  storage of partial results, and selecting maximal dimension cases are made available for polynomial systems through this map.  Conversely, polynomial features are made available for differential systems by using PDEToPolynomial. PDE solving can also benefit from this correspondence.

Applied to a single polynomial (or PDE), the function PolynomialToPDE (or PDEToPolynomial) is respectively:

Here $\mathrm{depvars}=\[u\]=\[\mathrm{u1},\mathrm{...},\mathrm{uk}\]$ are the dependent variables, and $\mathrm{p1},\mathrm{p2},...,\mathrm{pk}$ are polynomials in the independent variables $\mathrm{vars}=\[x\]=\[\mathrm{x1},\mathrm{...},\mathrm{xn}\]$. Also, $\mathrm{D1},\mathrm{D2},...,\mathrm{Dn}$ are the partial differential operators corresponding to $\mathrm{x1},\mathrm{x2},...,\mathrm{xn}$. So PolynomialToPDE applies to polynomials which are linear and homogeneous in $\mathrm{depvars}=\left[u\right]$, with coefficients that are polynomials in $\mathrm{vars}=\left[x\right]$.   PolynomialToPDE (or PDEToPolynomial) of a list is obtained by applying the functions described above to each member of the list.

PDEToPolynomial applies to PDEs which are linear and homogeneous in $\mathrm{depvars}=\left[u\right]$, with coefficients independent of $\mathrm{vars}=\left[x\right]$.

In the case of one dependent variable the dependent variable is omitted: PDEToPolynomial([p(D1,...,Dn) u1(x)], [x], [u1]) yields $p\left(\mathrm{x1},...,\mathrm{xn}\right)$. Similarly, it is possible to apply PolynomialToPDE to a polynomial in vars, without any dependent variable  present in depvars.

This mapping between linear homogeneous system of partial differential equations is an isomorphism between the differential ring and the corresponding polynomial ring (viewed as a module generated by $\left(\mathrm{u1},\mathrm{u2},...,\mathrm{uk}\right)$ ).

These functions are part of the PolynomialTools package, and so they can be used in the form PolynomialToPDE(..) only after executing the command with(PolynomialTools). However, they can always be accessed through the long form of the command by using PolynomialTools[PolynomialToPDE](..).

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

$S≔\mathrm{PolynomialToPDE}\left(\left[\left(x^2-2x+1\right)u+x^{3}v\right]\,\left[x\right]\,\left[u\,v\right]\right)$

$S≔\left[\frac{{\ⅆ}^3}{\ⅆx^3}v\left(x\right)+\frac{{\ⅆ}^2}{\ⅆx^2}u\left(x\right)-2\frac{\ⅆ}{\ⅆx}u\left(x\right)+u\left(x\right)\right]$

$\mathrm{PDEToPolynomial}\left(S\,\left[x\right]\,\left[u\,v\right]\right)$

$\left[\left(x^2-2x+1\right)u+x^{3}v\right]$

$\mathrm{PDEToPolynomial}\left(\left[2\mathrm{diff}\left(u\left(x\,y\right)\,x\,y\right)+au\left(x\,y\right)\right]\,\left[x\,y\right]\,\left[u\right]\right),\mathrm{PolynomialToPDE}\left(\left[2xy+a\right]\,\left[x\,y\right]\,\left[u\right]\right)$

$\left[2xy+a\right],\left[2\frac{{\partial }^2}{\partial x\partial y}u\left(x\,y\right)+au\left(x\,y\right)\right]$

$W≔\left[x^2-2xz+5\,x{y}^2+y{z}^3\,3y^2-8z^3\right]\:$

$\mathrm{Groebner}\left[\mathrm{Basis}\right]\left(W\,\mathrm{tdeg}\left(x\,y\,z\right)\right)$

$\left[x^2-2xz+5\,8z^3-3y^2\,8x{y}^2+3y^3\,9y^4+48y^{3}z+320y^2\right]$

$\mathrm{DW}≔\mathrm{PolynomialToPDE}\left(W\,\left[x\,y\,z\right]\,\left[u\right]\right)\:$$\mathrm{RDW}≔\mathrm{DEtools}\left[\mathrm{rifsimp}\right]\left(\mathrm{DW}\right)\:$

$\mathrm{PDEToPolynomial}\left(\mathrm{RDW}\left[\mathrm{Solved}\right]\,\left[x\,y\,z\right]\,\left[u\right]\right)$

$\left[\frac{y^2\left(9y^2+48yz+320\right)}{9}\,\frac{y^2\left(8x+3y\right)}{8}\,z^3-\frac{3y^2}{8}\,x^2-2xz+5\right]$

$S≔\left[-\mathrm{diff}\left(\mathrm{\ξ1}\left(x\,y\,z\right)\,x\right)+\mathrm{diff}\left(\mathrm{\ξ2}\left(x\,y\,z\right)\,y\right)\,-\mathrm{diff}\left(\mathrm{\ξ1}\left(x\,y\,z\right)\,y\right)-\mathrm{diff}\left(\mathrm{\ξ2}\left(x\,y\,z\right)\,x\right)\,-\mathrm{diff}\left(\mathrm{\ξ1}\left(x\,y\,z\right)\,z\right)-\mathrm{diff}\left(\mathrm{\ξ3}\left(x\,y\,z\right)\,x\right)\,-\mathrm{diff}\left(\mathrm{\ξ3}\left(x\,y\,z\right)\,y\right)-\mathrm{diff}\left(\mathrm{\ξ2}\left(x\,y\,z\right)\,z\right)\,-\mathrm{diff}\left(\mathrm{\ξ3}\left(x\,y\,z\right)\,z\right)+\mathrm{diff}\left(\mathrm{\ξ2}\left(x\,y\,z\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\,z\right)\,\mathrm{`\$`}\left(y\,2\right)\right)+\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\,z\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\,z\right)\,\mathrm{`\$`}\left(z\,2\right)\right)\,-\mathrm{diff}\left(\mathrm{\ξ2}\left(x\,y\,z\right)\,\mathrm{`\$`}\left(y\,2\right)\right)-\mathrm{diff}\left(\mathrm{\ξ2}\left(x\,y\,z\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+2\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\,z\right)\,y\right)-\mathrm{diff}\left(\mathrm{\ξ2}\left(x\,y\,z\right)\,\mathrm{`\$`}\left(z\,2\right)\right)\,-\mathrm{diff}\left(\mathrm{\ξ3}\left(x\,y\,z\right)\,\mathrm{`\$`}\left(x\,2\right)\right)-\mathrm{diff}\left(\mathrm{\ξ3}\left(x\,y\,z\right)\,\mathrm{`\$`}\left(y\,2\right)\right)-\mathrm{diff}\left(\mathrm{\ξ3}\left(x\,y\,z\right)\,\mathrm{`\$`}\left(z\,2\right)\right)+2\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\,z\right)\,z\right)\,-\mathrm{diff}\left(\mathrm{\ξ1}\left(x\,y\,z\right)\,\mathrm{`\$`}\left(z\,2\right)\right)+2\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\,z\right)\,x\right)-\mathrm{diff}\left(\mathrm{\ξ1}\left(x\,y\,z\right)\,\mathrm{`\$`}\left(y\,2\right)\right)-\mathrm{diff}\left(\mathrm{\ξ1}\left(x\,y\,z\right)\,\mathrm{`\$`}\left(x\,2\right)\right)\right]\:$

$\mathrm{with}\left(\mathrm{Ore\_algebra}\right)\:$

$\mathrm{PS}≔\mathrm{PDEToPolynomial}\left(S\,\left[x\,y\,z\right]\,\left[\mathrm{\ξ1}\,\mathrm{\ξ2}\,\mathrm{\ξ3}\,\mathrm{\eta }\right]\right)$

$\mathrm{PS}≔\left[-x\mathrm{\ξ1}+\mathrm{\ξ2}y\,-x\mathrm{\ξ2}-y\mathrm{\ξ1}\,-x\mathrm{\ξ3}-z\mathrm{\ξ1}\,-z\mathrm{\ξ2}-\mathrm{\ξ3}y\,\mathrm{\ξ2}y-\mathrm{\ξ3}z\,\mathrm{\eta }\left(x^2+y^2+z^2\right)\,\left(-x^2-y^2-z^2\right)\mathrm{\ξ2}+2\mathrm{\eta }y\,\left(-x^2-y^2-z^2\right)\mathrm{\ξ3}+2\mathrm{\eta }z\,\left(-x^2-y^2-z^2\right)\mathrm{\ξ1}+2\mathrm{\eta }x\right]$

$A≔\mathrm{Ore\_algebra}\left[\mathrm{poly\_algebra}\right]\left(x\,y\,z\,\mathrm{\ξ1}\,\mathrm{\ξ2}\,\mathrm{\ξ3}\,\mathrm{\eta }\right)\:$

$T≔\mathrm{Groebner}\left[\mathrm{MonomialOrder}\right]\left(A\,\mathrm{lexdeg}\left(\left[\mathrm{\eta }\,\mathrm{\ξ3}\,\mathrm{\ξ2}\,\mathrm{\ξ1}\right]\,\left[z\,x\,y\right]\right)\,\left[\mathrm{\eta }\,\mathrm{\ξ3}\,\mathrm{\ξ2}\,\mathrm{\ξ1}\right]\right)\:$

$\mathrm{GSG}≔\mathrm{Groebner}\left[\mathrm{Basis}\right]\left(\mathrm{PS}\,T\right)$

$\mathrm{GSG}≔\left[\mathrm{\ξ1}yz\,x^2\mathrm{\ξ1}+\mathrm{\ξ1}{y}^2\,-\mathrm{\ξ1}{y}^2+\mathrm{\ξ1}{z}^2\,\mathrm{\ξ1}{y}^3\,x\mathrm{\ξ1}{y}^2\,-x\mathrm{\ξ1}+\mathrm{\ξ2}y\,x\mathrm{\ξ2}+y\mathrm{\ξ1}\,x\mathrm{\ξ1}y+\mathrm{\ξ2}{z}^2\,z\mathrm{\ξ2}+\mathrm{\ξ3}y\,x\mathrm{\ξ3}+z\mathrm{\ξ1}\,-x\mathrm{\ξ1}+\mathrm{\ξ3}z\,x\mathrm{\ξ1}y+2\mathrm{\eta }y\,-\mathrm{\ξ1}{y}^2+2\mathrm{\eta }x\,xz\mathrm{\ξ1}+2\mathrm{\eta }z\right]$

$\mathrm{PolynomialToPDE}\left(\mathrm{GSG}\,\left[x\,y\,z\right]\,\left[\mathrm{\ξ1}\,\mathrm{\ξ2}\,\mathrm{\ξ3}\,\mathrm{\eta }\right]\right)$

$\left[\frac{{\partial }^2}{\partial y\partial z}\mathrm{\ξ1}\left(x\,y\,z\right)\,\frac{{\partial }^2}{\partial x^2}\mathrm{\ξ1}\left(x\,y\,z\right)+\frac{{\partial }^2}{\partial y^2}\mathrm{\ξ1}\left(x\,y\,z\right)\,-\frac{{\partial }^2}{\partial y^2}\mathrm{\ξ1}\left(x\,y\,z\right)+\frac{{\partial }^2}{\partial z^2}\mathrm{\ξ1}\left(x\,y\,z\right)\,\frac{{\partial }^3}{\partial y^3}\mathrm{\ξ1}\left(x\,y\,z\right)\,\frac{{\partial }^3}{\partial x\partial y^2}\mathrm{\ξ1}\left(x\,y\,z\right)\,-\frac{\partial }{\partial x}\mathrm{\ξ1}\left(x\,y\,z\right)+\frac{\partial }{\partial y}\mathrm{\ξ2}\left(x\,y\,z\right)\,\frac{\partial }{\partial x}\mathrm{\ξ2}\left(x\,y\,z\right)+\frac{\partial }{\partial y}\mathrm{\ξ1}\left(x\,y\,z\right)\,\frac{{\partial }^2}{\partial x\partial y}\mathrm{\ξ1}\left(x\,y\,z\right)+\frac{{\partial }^2}{\partial z^2}\mathrm{\ξ2}\left(x\,y\,z\right)\,\frac{\partial }{\partial z}\mathrm{\ξ2}\left(x\,y\,z\right)+\frac{\partial }{\partial y}\mathrm{\ξ3}\left(x\,y\,z\right)\,\frac{\partial }{\partial x}\mathrm{\ξ3}\left(x\,y\,z\right)+\frac{\partial }{\partial z}\mathrm{\ξ1}\left(x\,y\,z\right)\,-\frac{\partial }{\partial x}\mathrm{\ξ1}\left(x\,y\,z\right)+\frac{\partial }{\partial z}\mathrm{\ξ3}\left(x\,y\,z\right)\,\frac{{\partial }^2}{\partial x\partial y}\mathrm{\ξ1}\left(x\,y\,z\right)+2\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\,z\right)\,-\frac{{\partial }^2}{\partial y^2}\mathrm{\ξ1}\left(x\,y\,z\right)+2\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\,z\right)\,\frac{{\partial }^2}{\partial x\partial z}\mathrm{\ξ1}\left(x\,y\,z\right)+2\frac{\partial }{\partial z}\mathrm{\eta }\left(x\,y\,z\right)\right]$

$\mathrm{RS}≔\mathrm{DEtools}\left[\mathrm{rifsimp}\right]\left(S\,\left[\mathrm{\eta }\,\mathrm{\ξ3}\,\mathrm{\ξ2}\,\mathrm{\ξ1}\right]\right)$

$\mathrm{RS}≔table\left(\left[\mathrm{Solved}=\left[\frac{{\partial }^2}{\partial x^2}\mathrm{\eta }\left(x\,y\,z\right)=0\,\frac{{\partial }^2}{\partial x\partial y}\mathrm{\eta }\left(x\,y\,z\right)=0\,\frac{{\partial }^2}{\partial x\partial z}\mathrm{\eta }\left(x\,y\,z\right)=0\,\frac{{\partial }^2}{\partial y^2}\mathrm{\eta }\left(x\,y\,z\right)=0\,\frac{{\partial }^2}{\partial y^2}\mathrm{\ξ1}\left(x\,y\,z\right)=2\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\,z\right)\,\frac{{\partial }^2}{\partial y\partial z}\mathrm{\eta }\left(x\,y\,z\right)=0\,\frac{{\partial }^2}{\partial y\partial z}\mathrm{\ξ1}\left(x\,y\,z\right)=0\,\frac{{\partial }^2}{\partial z^2}\mathrm{\eta }\left(x\,y\,z\right)=0\,\frac{{\partial }^2}{\partial z^2}\mathrm{\ξ3}\left(x\,y\,z\right)=-2\frac{\partial }{\partial z}\mathrm{\eta }\left(x\,y\,z\right)\,\frac{{\partial }^2}{\partial z^2}\mathrm{\ξ2}\left(x\,y\,z\right)=2\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\,z\right)\,\frac{{\partial }^2}{\partial z^2}\mathrm{\ξ1}\left(x\,y\,z\right)=2\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\,z\right)\,\frac{\partial }{\partial x}\mathrm{\ξ3}\left(x\,y\,z\right)=-\frac{\partial }{\partial z}\mathrm{\ξ1}\left(x\,y\,z\right)\,\frac{\partial }{\partial x}\mathrm{\ξ2}\left(x\,y\,z\right)=-\frac{\partial }{\partial y}\mathrm{\ξ1}\left(x\,y\,z\right)\,\frac{\partial }{\partial x}\mathrm{\ξ1}\left(x\,y\,z\right)=\frac{\partial }{\partial z}\mathrm{\ξ3}\left(x\,y\,z\right)\,\frac{\partial }{\partial y}\mathrm{\ξ3}\left(x\,y\,z\right)=-\frac{\partial }{\partial z}\mathrm{\ξ2}\left(x\,y\,z\right)\,\frac{\partial }{\partial y}\mathrm{\ξ2}\left(x\,y\,z\right)=\frac{\partial }{\partial z}\mathrm{\ξ3}\left(x\,y\,z\right)\right]\right]\right)$

$\mathrm{sys}≔\left[u+av+a^{2}w\,u+bv+b^{2}w\,u+cv+c^{2}w\,u+cv+\left(c^2+c-2\right)w\right]\:$

$\mathrm{Dsys}≔\mathrm{PolynomialToPDE}\left(\mathrm{sys}\,\left[a\right]\,\left[u\,v\,w\right]\right)\:$

$\mathrm{RDsys}≔\mathrm{DEtools}\left[\mathrm{rifsimp}\right]\left(\mathrm{Dsys}\,\mathrm{casesplit}\right)$

$\mathrm{RDsys}≔table\left(\left[1=table\left(\left[\mathrm{Pivots}=\left[-b+c\ne 0\,c-2\ne 0\right]\,\mathrm{Case}=\left[\left[-b+c\ne 0\,v\left(a\right)\right]\,\left[c-2\ne 0\,w\left(a\right)\right]\right]\,\mathrm{Solved}=\left[u\left(a\right)=0\,v\left(a\right)=0\,w\left(a\right)=0\right]\right]\right)\,2=table\left(\left[\mathrm{Pivots}=\left[-b+2\ne 0\right]\,\mathrm{Case}=\left[\left[-b+c\ne 0\,v\left(a\right)\right]\,\left[c-2=0\,w\left(a\right)\right]\right]\,\mathrm{Solved}=\left[\frac{{\ⅆ}^2}{\ⅆa^2}w\left(a\right)=-2bw\left(a\right)+\left(\frac{\ⅆ}{\ⅆa}w\left(a\right)\right)b+2\frac{\ⅆ}{\ⅆa}w\left(a\right)\,u\left(a\right)=2bw\left(a\right)\,v\left(a\right)=-w\left(a\right)\left(b+2\right)\,c=2\right]\right]\right)\,3=table\left(\left[\mathrm{Pivots}=\left[c-2\ne 0\right]\,\mathrm{Case}=\left[\left[-b+c=0\,v\left(a\right)\right]\,\left[c-2\ne 0\,w\left(a\right)\right]\right]\,\mathrm{Solved}=\left[\frac{\ⅆ}{\ⅆa}v\left(a\right)=cv\left(a\right)\,u\left(a\right)=-cv\left(a\right)\,w\left(a\right)=0\,b=c\right]\right]\right)\,4=table\left(\left[\mathrm{Case}=\left[\left[-b+c=0\,v\left(a\right)\right]\,\left[c-2=0\,w\left(a\right)\right]\right]\,\mathrm{Solved}=\left[\frac{{\ⅆ}^2}{\ⅆa^2}w\left(a\right)=-\frac{\ⅆ}{\ⅆa}v\left(a\right)+4w\left(a\right)+2v\left(a\right)\,u\left(a\right)=-4w\left(a\right)-2v\left(a\right)\,b=2\,c=2\right]\right]\right)\,\mathrm{casecount}=4\right]\right)$