Differential polynomial

A differential polynomial in a smooth vector-valued unknown $u=(u_1,\ldots,u_m)$ is a finite sum of expressions

c(x)\prod_{j=1}^{N}\partial^{I_j}u_{a_j}(x),

where the coefficients $c$ are prescribed smooth functions, $I_j$ are multi-indices, and $a_j$ specify components. The order is the largest derivative order $|I_j|$ used; the degree is the largest number $N$ of factors in a monomial. Degree-zero terms depend only on the coefficients.

Different order and degree

The expression $u\,\partial_xu+\partial_x^2u$ has differential order two and polynomial degree two. Substituting formal series determines each coefficient through finitely many terms because the order and degree are finite. A composition such as $e^u$ is not a differential polynomial in $u$ .

Residual comparison

The difference estimate bounds a nonlinear change by the input difference jet. Combined with increasing-order tails, it gives flat residual realization.