# Transforming an inequality in another variables via a polynomial relation

Given an inequality through a quadratic polynomial

$$ax_1^2+bx_2^2 +cx_1x_3+dx_2x_4\leq 0,$$

where $$c_i\in \mathrm{R}$$, $$0\leq x_i \leq 1$$, $$i=1,2,3,4$$, can it be somehow transformed into an inequality in terms of $$y_i$$, $$i=1,2,3,4$$, where

$$\sum_{i=1}^4 x_i^1 = y_1, \sum_{i=1}^4 x_i^2 = y_2, \sum_{i=1}^4 x_i^3 = y_3, \sum_{i=1}^4 x_i^4 = y_4$$

establishes the relation between $$x_i$$ and $$y_i$$.

How does on go from a function $$f(x_1,x_2,x_3,x_4)$$ over to a function $$g(y_1,y_2,y_3,y_4)$$? Is such a transformation possible? If so, in what direction proceed to prove its existence? How to explicitly construct such a transformation?

I am not even sure how to start, any hints on improving this question welcome.

• I can't see how powers of 3 or 4 can play any role. You also have not written down any inequalities involving your sums so you may need to clarify your question.. – Paul Oct 4 '18 at 14:14
• @Paul The powers of 3 and 4 define $y_3$ and $y_4$. We have 4 equations of 4 variables which gives us the transformation. – wondering Oct 4 '18 at 14:46
• @wondering The $y_j$ are symmetric in $x_i$ while the inequality is not. So short answer is no, there is no such transformation (unless of course you have additional conditions). – dxiv Oct 4 '18 at 16:55