= Solution
Under spatial inversion the <electric polarization>, a <polar vector>, changes from $\mathbf P$ to $-\mathbf P$. <Neumann's principle> requires an equilibrium property of a phase retaining <inversion symmetry> to be unchanged. Thus $\mathbf P=-\mathbf P$, forcing $\mathbf P=0$: a centrosymmetric phase cannot itself be <ferroelectric>. A transition can, of course, remove that <symmetry>.
For an inversion-symmetric parent phase the <Landau theory> <free energy> satisfies $F(\mathbf P)=F(-\mathbf P)$. Every polynomial term of odd total degree changes sign and must have zero coefficient. Restricting to one fixed direction gives an even scalar expansion in $P$.
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