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Use coordinates in which the current spacing is $h$. On the first child subgrid, the polynomial candidate ison the second it isFor arbitrary sufficiently long uniformly spaced data, all refined vertices can lie on one degree-$n$ polynomial only if these two candidate polynomials agree. They agree with $P$ when $n=0,1$. For $P(x)=a{x}^2+bx+c$, expansion givesThus quadratics preserve their degree, although not the same values.

For a polynomial of degree $n\ge 3$, subtract the two finite Taylor expansions. The first-derivative term cancels, while the leading nonzero differential term isIf the leading coefficient of $P$ is $a_n\ne 0$, this difference has nonzero coefficient $a_{n}n(n-1)(n-2)h^3/32$ at degree $n-3$; the higher derivatives have lower degree and cannot cancel it. Therefore the polynomial degree preservation under Chaikin subdivision holds exactly forThis is a general refinement property, independent of accidental interpolation through a small finite number of vertices.