 Polynomial model of a regularity structure

The polynomial model of a regularity structure realizes abstract monomials as ordinary coordinate differences around each base point. It is the example that turns a regularity structure into familiar Taylor polynomials. In $d$ dimensions, use a polynomial ring in symbols $X_1,\dots ,X_d$, graded by total degree. For a multi-index $k=(k_1,\dots ,k_d)$,Re-expansion uses ${\Gamma }_{xy}p(X)=p(X+x-y)$. Substitution immediately gives ${\Pi }_x{\Gamma }_{xy}={\Pi }_y$ and ${\Gamma }_{xy}{\Gamma }_{yz}={\Gamma }_{xz}$, the compatibility identities of a model of a regularity structure.^[ref]

For example, $f(z_1,z_2)=z_1{z}_2$ has local abstract expansion $a_x=2+2X_1+X_2+X_1{X}_2$ at $x=(1,2)$. At $y=(3,4)$ it has $a_y=12+4X_1+3X_2+X_1{X}_2$. Transporting the latter substitutes $X_1-2,X_2-2$:The coefficients change because the expansion point changes; the represented function is still $z_1{z}_2$.

The analytic scaling also follows by calculation. For a smooth test function and ${\phi }_x^{\lambda }(z)={\lambda }^{-d}\phi ((z-x)/\lambda )$, changing variables givesThus the assigned degree predicts exactly the power of $\lambda$ in the model bound. Anisotropic scaling can assign different weights to different coordinates instead; ordinary total degree is the isotropic choice. General regularity structures retain this organization of local expansions while admitting symbols that are not polynomials.