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The Bourgain-Demeter decoupling theorem controls a function whose Fourier support lies near a positively curved hypersurface by the contributions from small frequency caps. For a compact $C^2$ hypersurface in ${\mathbb{R}}^n$, $n\ge 2$, with positive-definite second fundamental form, decompose its $\delta$-neighborhood into caps of tangential width ${\delta }^{1/2}$ and normal thickness $\delta$. If $f_{\theta }$ denotes the corresponding Fourier pieces, the critical estimate isThe constant is independent of $\delta \in (0,1)$ and $f$, for the fixed surface and chosen cap decomposition. The same arbitrarily small loss holds for $2\le p\le p_c$; supercritical exponents require an additional scale power.^[ref]

For a parabola in ${\mathbb{R}}^2$, $p_c=6$. A naive triangle inequality sums all cap norms, whereas decoupling sums their squares before taking a square root. Curvature makes this much stronger control possible. The theorem links Fourier restriction theory with estimates for oscillatory sums and dispersive equations.

The degree-$d$ moment curve is $\gamma (t)=(t,t^2,\dots ,t^d)$ in ${\mathbb{R}}^d$. Its tangent vectors at distinct parameters are linearly independent:Factor $1,2,\dots ,d$ from the rows and apply the Vandermonde determinant formula. Parameter separation bounded below therefore gives quantitative transversality of the tangent directions.

For radial position vectors, the determinant instead equals $({\prod }_j{t}_j){\prod }_{i<j}(t_j-t_i)$. They are independent only when all parameters are nonzero as well as distinct. In particular $\gamma (0)=0$ supplies no radial direction. For $d=2$, the curve is the parabola, with tangent determinant $2(t_2-t_1)$. Its nondegeneracy is a basic source of curvature in Fourier restriction theory and decoupling inequalities.