= Solution
Take $D_j=-i\partial_{x_j}$ and the <Fourier transform> convention $\widehat f(\lambda)=\int e^{-ix\cdot\lambda}f(x)\,dx$. Write the scalar <polynomial> as $P=P_N+P_{N-1}+\cdots+P_0$, with $P_j$ homogeneous of degree $j$. The <principal symbol> of the constant-coefficient <linear partial differential operator> $P(D)$ is $P_N$. It is an <elliptic differential operator> exactly when
$$
\boxed{P_N(\lambda)\ne0\quad\text{for every real }\lambda\ne0.}
$$
Complex coefficients are allowed, but the frequency $\lambda$ is real.
By compactness of the unit <sphere>, $c=\min_{|\omega|=1}|P_N(\omega)|>0$. Homogeneity gives $|P_N(\lambda)|\geq c|\lambda|^N$, whereas the lower-degree terms are bounded by $C|\lambda|^{N-1}$ for $|\lambda|\geq1$. The <triangle inequality> then gives the <high-frequency lower bound for an elliptic polynomial>:
$$
|P(\lambda)|\geq c|\lambda|^N-C|\lambda|^{N-1}
\geq\frac c2|\lambda|^N\geq c'\langle\lambda\rangle^N
$$
for sufficiently large $|\lambda|$. Thus
$$
\boxed{|P(\lambda)|\gtrsim\langle\lambda\rangle^N\quad(|\lambda|\geq R).}
$$
The <Japanese bracket> is $\langle\lambda\rangle=(1+|\lambda|^2)^{1/2}$. If $N=0$, a nonzero constant $P$ satisfies the same assertion directly.
For real $s$, the <Sobolev space> is the <tempered distribution> space
$$
H^s(\mathbb R^n)=\left\{u\in\mathcal S'(\mathbb R^n):
\widehat u\text{ is a function and }
\int_{\mathbb R^n}\langle\lambda\rangle^{2s}|\widehat u(\lambda)|^2\,d\lambda<\infty\right\}.
$$
One may include $(2\pi)^{-n}$ in the squared norm for this <Fourier transform> normalization; it does not change the space. The <Local Sobolev space> is
$$
H^s_{\mathrm{loc}}(X)=\{u\in\mathcal D'(X):\chi u\in H^s(\mathbb R^n)
\text{ for every }\chi\in C_c^\infty(X)\},
$$
where <multiplication of a distribution by a smooth function> is followed by extension by zero.
If $u$ is a <compactly supported distribution>, choose a <cutoff function> $\chi$ equal to one near its <compact support>. The finite <order of a distribution> gives an integer $M$ and a bound
$$
|\langle u,\psi\rangle|\leq C\max_{|\alpha|\leq M}
\sup_{x\in K}|\partial^\alpha\psi(x)|
$$
for a fixed <compact set> $K$. This also makes $u$ a <tempered distribution>. Its <Fourier transform of a compactly supported distribution> is the <smooth function>
$$
\widehat u(\lambda)=\langle u,\chi(x)e^{-ix\cdot\lambda}\rangle,
\qquad |\widehat u(\lambda)|\leq C'\langle\lambda\rangle^M.
$$
The <Leibniz rule> gives the last estimate, and parameter differentiation under the finite-order pairing proves smoothness of this <Fourier transform>. Therefore the <negative Sobolev regularity of a compactly supported distribution> is
$$
\boxed{u\in H^s(\mathbb R^n)\quad\text{for every }s<-M-\frac n2.}
$$
Indeed $\langle\lambda\rangle^{2(s+M)}$ is integrable precisely when $2(s+M)<-n$. This proves the claimed existence of a sufficiently negative <Sobolev space> index.
We use three elementary <Sobolev space> facts: differentiation of order $q$ maps $H^t$ continuously into $H^{t-q}$; $H^a\subseteq H^b$ for $a\geq b$; and <Sobolev multiplication by a smooth cutoff> is bounded on $H^t$ for every real $t$. The first two follow directly from the frequency weights. For the third, <multiplication of a distribution by a smooth function> becomes convolution with the rapidly decaying $\widehat\chi$, and
$$
\langle\lambda\rangle^t\leq C_t\langle\lambda-\eta\rangle^{|t|}\langle\eta\rangle^t
$$
together with <Young's convolution inequality> gives the bound. These facts apply after localization as well.
The <high-frequency lower bound for an elliptic polynomial> gives a useful global implication. If $w\in H^r(\mathbb R^n)$ and $P(D)w\in H^t(\mathbb R^n)$, split its <Fourier transform> into $|\lambda|\leq R$ and $|\lambda|>R$. The low-frequency part is controlled by $\|w\|_{H^r}$, while the high-frequency part is controlled by $\|P(D)w\|_{H^t}$. Thus, for arbitrary real $r,t$,
$$
\boxed{\|w\|_{H^{t+N}}\leq C_{r,t}\left(\|P(D)w\|_{H^t}+\|w\|_{H^r}\right).}
$$
The conclusion that $w\in H^{t+N}$ is obtained directly by integrating these frequency bounds; it is not an assumption made to state the estimate.
For the <cutoff bootstrap for local elliptic regularity>, fix $U\Subset X$. A <cutoff function> equal to one near $\overline U$ makes $u$ a <compactly supported distribution> after multiplication, so the preceding negative-index argument gives $u\in H^r_{\mathrm{loc}}(U)$ for some finite $r$. Suppose inductively that $u\in H^q_{\mathrm{loc}}(U)$. For any <test function> $\chi\in C_c^\infty(U)$,
$$
P(D)(\chi u)=\chi P(D)u+[P(D),\chi]u.
$$
The <Leibniz rule> shows that this <commutator> has order at most $N-1$, with <smooth functions> as coefficients, all with <compact support>:
$$
[P(D),\chi]=\sum_{|\alpha|\leq N}p_\alpha
\sum_{0<\beta\leq\alpha}\binom\alpha\beta
(D^\beta\chi)D^{\alpha-\beta}.
$$
A second <cutoff function> equal to one near $\operatorname{supp}\chi$ lets us apply the stated <Sobolev space> bounds to every term. Hence $[P(D),\chi]u\in H^{q-N+1}(\mathbb R^n)$, while $\chi P(D)u\in H^s(\mathbb R^n)$. For $N\geq1$, the global estimate yields
$$
\chi u\in H^{\min(s,q-N+1)+N}
=H^{\min(s+N,q+1)}.
$$
This holds for every such $\chi$, so it improves the <Local Sobolev space> index by one until $s+N$ is reached. A finite number of iterations starting at $r$ proves
$$
\boxed{P(D)u\in H^s_{\mathrm{loc}}(X)\quad\Longrightarrow\quad
u\in H^{s+N}_{\mathrm{loc}}(X).}
$$
Since $U\Subset X$ was arbitrary, the conclusion holds throughout $X$. For $N=0$, division by the nonzero constant $P$ proves it immediately. This is the asserted <elliptic regularity>, proved without assuming an initial nonnegative <Sobolev space> index.
A first-order example on $\mathbb R^2$ is the <first-order Cauchy-Riemann operator>
$$
\boxed{P(D)=D_1+iD_2=-2i\,\partial_{\bar z},\qquad
P_1(\lambda)=\lambda_1+i\lambda_2,\qquad |P_1(\lambda)|=|\lambda|.}
$$
It is an <elliptic differential operator> with complex coefficients. In one real variable, $D=-i\,d/dx$ is already a first-order <elliptic differential operator>.
\b[There is no scalar <elliptic differential operator> of odd order in three variables.] If its order $N$ were odd, its <principal symbol> would satisfy $P_N(-\omega)=-P_N(\omega)$ on $S^2$. Regard the <continuous map>
$$
f:S^2\longrightarrow\mathbb R^2,\qquad
f(\omega)=(\operatorname{Re}P_N(\omega),\operatorname{Im}P_N(\omega)).
$$
The <Borsuk-Ulam theorem> gives $f(\omega)=f(-\omega)$ for some $\omega$. Oddness then gives $f(\omega)=0$, contradicting ellipticity. This is the <odd-order obstruction for scalar elliptic operators in at least three dimensions>; restricting to a three-dimensional subspace proves the higher-dimensional case too. For real coefficients alone, the same obstruction follows from the <intermediate value theorem> along a path between antipodal points.
The scalar qualification matters. An <elliptic system of differential equations> can be first order in three variables: with the <Pauli matrices>, the matrix symbol $A(\lambda)=\sum_{j=1}^3\sigma_j\lambda_j$ satisfies $A(\lambda)^2=|\lambda|^2\mathbf1$ and is invertible for $\lambda\ne0$. This does not contradict the scalar <polynomial> obstruction.
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