Given an integer $n \geqslant 3$ the polynomial $f(x_1, \ldots, x_n)$ with integer coefficients is called [i]good[/i] if $f(0,\ldots, 0) = 0$ and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of $\pi$ of the numbers $1,\ldots, n$. Denote by $\mathcal{J}$ the set of polynomials of the form \[p_1q_1+\cdots+p_mq_m,\]where $m$ is a positive integer and $q_1,\ldots , q_m$ are polynomials with integer coefficients, and $p_1,\ldots , p_m$ are good polynomials. Find the smallest natural number $D{}$ such that each monomial of degree $D{}$ lies in the set $\mathcal{J}$.
1. **Understanding the Problem:**
   We need to find the smallest natural number \( D \) such that every monomial of degree \( D \) can be expressed as a sum of products of good polynomials and other polynomials with integer coefficients.

2. **Properties of Good Polynomials:**
   A polynomial \( f(x_1, \ldots, x_n) \) is called good if:
   - \( f(0, \ldots, 0) = 0 \)
   - \( f(x_1, \ldots, x_n) = f(x_{\pi_1}, \ldots, x_{\pi_n}) \) for any permutation \( \pi \) of the numbers \( 1, \ldots, n \).

3. **Form of Good Polynomials:**
   Good polynomials are symmetric and vanish at the origin. Examples include:
   - \( x_1 + x_2 + \cdots + x_n \)
   - \( x_1^2 + x_2^2 + \cdots + x_n^2 \)
   - \( x_1 x_2 + x_1 x_3 + \cdots + x_{n-1} x_n \)

4. **Constructing Monomials:**
   We need to express any monomial of degree \( D \) in terms of sums of products of good polynomials and other polynomials with integer coefficients.

5. **Degree Analysis:**
   - For \( n = 3 \), consider the monomial \( x_1^D \).
   - We need to express \( x_1^D \) using good polynomials.

6. **Symmetric Polynomials:**
   Symmetric polynomials of degree \( D \) can be written as sums of products of elementary symmetric polynomials. For example, for \( n = 3 \):
   - \( e_1 = x_1 + x_2 + x_3 \)
   - \( e_2 = x_1 x_2 + x_1 x_3 + x_2 x_3 \)
   - \( e_3 = x_1 x_2 x_3 \)

7. **Degree \( D = 3 \):**
   - Consider \( x_1^3 \). We can write:
     \[
     x_1^3 = \frac{1}{6} \left( (x_1 + x_2 + x_3)^3 - 3(x_1 + x_2 + x_3)(x_1 x_2 + x_1 x_3 + x_2 x_3) + 3x_1 x_2 x_3 \right)
     \]
   - This expression uses symmetric polynomials, which are good polynomials.

8. **Generalization:**
   - For any \( n \geq 3 \), the same approach can be used to express any monomial of degree \( D \) using symmetric polynomials.
   - The smallest \( D \) such that every monomial of degree \( D \) can be expressed in this way is \( D = 3 \).

Conclusion:
\[
\boxed{3}
\]