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I have the following complete homogeneous polynomial of degree $r$:

$p_(x_1, x_2,...x_n) = \sum_{i_1 + i_2 + ... +i_n = r, i_k\in {0,1,..r}} \phi_{i_1}(x_2)\phi_{i_2}(x_2)...\phi_{i_n}(x_n) $

where the functions $\phi$ have the form $\phi_{j} (x_k) =a_{jk}x_k^j$ and the $a_{jk}$ are real positive numbers, that is $a_{jk}>0$. Specifically we set $a_{0k}=1$ $ \forall k$. Therefore the above polynomial is complete in the sense the all the combinations of the $i_k$ are included in it. In particular if all the $a_{jk}$ are one, we have the complete symmetric polynomial. The polynomial is then defined by $(r+1)\times n$ coefficients. I have the following question: I define the following derivatives:

$D_l(i_1,i_2,..i_l) = \frac{\partial^l log(p) }{\partial x_{i_1} \partial x_{i_2} ... \partial x_{i_l}}$

where all the subscripts (the variables involved) $i_1, i_2,...i_l$ are different and $l\leq n$. Then my question is: Can we obtain any set of conditions for the coeffcients $a_{jk}$ so that the following inequalities hold for positive variables $x_1,x_2,..x_n$:

$D_l(i_1,i_2) <0$ for all $(i_1,i_2) \in \{1,..n\}^2$.

$D_l(i_1,i_2,i_3) >0$ for all $(i_1,i_2,i_3) \in \{1,..n\}^3$.

$D_l(i_1,i_2,i_3,i_4) <0$ for all $(i_1,i_2,i_3,i_4) \in \{1,..n\}^4$.

.... and finally

$(-1)^n D_n=(-1)^nD_n(1,2,3,,...n) <0$ .

Note that $D_l(i_1) >0$ trivially.

Any kind of help with this problem would be acknowledged. Another simpler version of this problem arises from making the coefficients dependent of $n$ parameters $b_k, k=1,...n$. for example we could define $a_{1k} =b_k$ and $a_{jk} =a_{j-1,k}\times j$ or something similar.


The above problem is equivalent to the following: find a multivariate discrete discrete distribution $(I_1,..I_n)$ whose support is the set of nonnegative integers $i_k\in {0,1,..r}$ with $i_1 + i_2 + ... +i_n = r$ and which is given by:

$Prob(I_1=i_1,...I_n=i_n) = \frac{ \phi_{i_1}(x_2)\phi_{i_2}(x_2)...\phi_{i_n}(x_n) }{p(x_1,...x_n)} = \frac{ a_{i_11}x_1^{i_1} a_{i_22}x_2^{i_2} ...a_{i_nn}x_n^{i_n} }{p(x_1,...x_n)} (*)$

Then the above conditions on the derivative the logarithm of the polynomial are equivalent to:

$E[H_{j_1} H_{j_2} ] <0$,

$E[H_{j_1} H_{j_2} H_{j_3} ] >0$,

$E[H_{j_1} H_{j_2} H_{j_3} H_{j_4} ] <0$e .... $(-1)^n E[H_{1} H_{2} ..H_{n}] <0$, with all the subscripts $j_k$ being different

and where $H_{j_k}=I_{j_k}-E[I_{j_k}]$ is the mean centered variable. This equivalence arises from the following consideration: the distribution given by (*) is an exponential familiy distribution in the "parameters" $x_1,...x_n$ and the polynomial $p(x_i,...x_n)$ is its normalizing constant (the constant that makes the probability sum to one). Then, the derivatives $D_l(i_1,i_2,..i_l)$ give the above expections.

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