This paper initializes the study of the Pierce-Birkhoff conjecture. We start by introducing the notion of the area and volume induced by a multivariate expansion and develop some inequalities for our next studies. In particular we obtain the inequality \begin{align} \sum \limits_{\substack{i,j\in [1,n]\\a_{i_{\sigma(s)}}<a_{j_{\sigma(s)}}\\s\in [1,l]\\v\neq i,j\\v\in [1,n] }}\bigg | \bigg |\vec{a}_{i} \diamond \vec{a}_{j}\diamond \cdots \diamond \vec{a}_v\bigg |\bigg |\sum \limits_{k=1}^{n}\int \limits_{a_{i_{\sigma(l)}}}^{a_{j_{\sigma(l)}}}\int \limits_{a_{i_{\sigma(l-1)}}}^{a_{j_{\sigma(l-1)}}}\cdots \int \limits_{a_{i_{\sigma(1)}}}^{a_{j_{\sigma(1)}}}g_kdx_{\sigma(1)}dx_{\sigma(2)}\cdots dx_{\sigma(l)}\nonumber\\ \leq 2C\times \binom{n}{2}\times \sqrt{n}\times \nonumber \\ \times \int \limits_{a_{i_{\sigma(l)}}}^{a_{j_{\sigma(l)}}}\int \limits_{a_{i_{\sigma(l-1)}}}^{a_{j_{\sigma(l-1)}}}\cdots \int \limits_{a_{i_{\sigma(1)}}}^{a_{j_{\sigma(1)}}}\sqrt{\bigg(\sum \limits_{k=1}^{n}(\mathrm{max}(g_k))^2\bigg)}dx_{\sigma(1)}dx_{\sigma(2)}\cdots dx_{\sigma(l)}\nonumber \end{align}for some constant $C>0$, where $\sigma:\{1,2,\ldots,l\}\longrightarrow \{1,2,\ldots,l\}$ is a permutation for $g_k\in \mathbb{R}[x_1,x_2,\ldots,x_l]$ and $\vec{a}_{i} \diamond \vec{a}_{j}\diamond \cdots \diamond \vec{a}_k \diamond \vec{a}_{v}$ is the cross product of any of the $n-1$ fixed spots in $\mathbb{R}^{l}$ including the spots $\vec{a}_i,\vec{a}_j$.