The author considers the classical problem of homogenization in the calculus of variations \[ f_{\hom}(\xi)= \inf\Biggl\{\int_Y f(x,Du(x)) dx\mid u\in W^{1,p}_{\text{loc}}(\mathbb{R}^n), Du\text{ is }Y\text{-periodic}, \langle Du\rangle= \xi\Biggr\}, \] where \(Y\) denotes the unit cube in \(\mathbb{R}^n\) and \(\langle\cdots\rangle\) denotes the average on \(Y\). The integrand is a function \(f:\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}\), which is, as usual, assumed to be measurable and \(Y\)-periodic with respect to the first variable, convex in the second variable, and, finally, to satisfy the \(p\)-growth condition \[ C_1|\xi|^p\leq f(x,\xi)\leq C_2(1+ |\xi|^p), \] with \(1 0\) for every \(\xi\in\mathbb{R}^n\) and almost every \(x\in \mathbb{R}^n\). In this paper the integrand is given by the homogeneous polynomial \[ f(x,\xi)= a(x)|\xi|^4,\quad \forall x\in \mathbb{R}^2,\quad \forall\xi\in \mathbb{R}^2, \] where \(a(x)= \alpha\chi(x_1)+ \beta(1-\chi(x_1))\), with \(0<\alpha< \beta<+\infty\), and \(\chi(t)\) denotes the 1-periodic extension to \(\mathbb{R}\) of the function \[ \chi_0(t)= \begin{cases} 1,\quad\text{if }0\leq t<\theta,\\ 0,\quad\text{if }\theta\leq t< 1,\end{cases} \] with \(0<\theta< 1\). The author proves that the corresponding \(f_{\hom}\) is not a polynomial.