Let $f\_1,\ldots,f\_r \in \mathbb{R}[x\_1,\ldots,x\_n]$ be $r$ homogeneous polynomials of the same odd degree $d$, where $d \in \{3,5,7,\ldots\}$.



> 
> For which values of $r,n,d$ there exists a **real** solution $a=(a\_1,\ldots,a\_n) \in \mathbb{R}^n-\{(0,\ldots,0)\}$
>  to the system of $r$ inequalities $f\_i \geq 0$?
> 
> 
> 


For $r=2$ the answer is positive, namely, there exists such $a$, by the following argument that can be found in the answer to [this question](https://math.stackexchange.com/questions/2847727/existence-of-a-real-solution-to-a-certain-system-of-inequalities/2847758#2847758): By continuity there exists $t \in \mathbb{R}^n$ with $|t| = 1$ and $f\_1(t) = f\_1(-t) = 0$. Clearly, at least one of $\{f\_2(t),f\_2(-t)\}$ is $\geq 0$. Then for $s=t$ or $s=-t$, we have: $f\_1(s)=0$ and $f\_2(s) \geq 0$.


Actually, I am interested in the case $n=4$; I really apologize that I am not explaining why (it is quite complicated; hopefully later I will be able to explain). 


**Remarks:** 


**(1)** I wonder if the following idea may yield an answer to
my question: For example, $n=4,d=3$, and order the monomials:
$y\_1:=x\_1^3,y\_2:=x\_1^2x\_2,y\_3:=x\_1^2x\_3,y\_4:=x\_1^2x\_4,x\_1x\_2x\_1,\ldots,y\_m:=x\_4^3$. Then the system of $r$ inequalities $f\_i \geq 0$ becomes a *linear* system in $m$ new variables $y\_1,\ldots,y\_m$. Now, $0=(0,\ldots,0) \in \mathbb{R}^m$ is a solution, and if the associated matrix is invertible, then $0$ is a unique solution, but this does not imply that the original system has $0 \in \mathbb{R}^n$ as a unique solution, so there is hope to get non-trivial solutions. On the other hand, if the associated matrix is not invertible, then there are non-trivial solutions, but I am not sure if it is possible to obtain from one of them a solution to the original system, probably not necessarily.


**(2)** [This paper](https://www.jstor.org/stable/1967869?seq=1#page_scan_tab_contents) seems relevant.


**(3)** I have already asked the above question [here](https://math.stackexchange.com/questions/2848559/solutions-to-a-system-of-homogeneous-equations-inequalities?noredirect=1&lq=1), but have not received any comments, neither to it nor to another [similar question](https://math.stackexchange.com/questions/2847838/existence-of-a-real-solution-to-a-certain-system-of-inequalities-generalized?noredirect=1&lq=1).


Any hints and comments are welcome!


**Edit:** I have now found [this question](https://mathoverflow.net/questions/265286/polynomial-equations-in-many-variables-have-solutions-lang-1952-paper?rq=1) which almost answers my question, provided that I understood it correctly and if we take $k=0$
in that question notations, then we get that my question, if we replace $\mathbb{R}$ by $\mathbb{C}$, has a positive answer (= a common zero for all the $f\_i$'s) for $n > r$.



Solutions to a system of homogeneous equations (inequalities)
I cannot say much about $\geq0$. For $=0$ one can prove the following statement: if $r\leq n-1$ then then there is an $a\in \mathbf R^n\setminus\{0\}$ such that $f\_i(a)=0$ for all $i$.


Indeed, each polynomial $f\_i$ defines a vanishing set $Z(f\_i)$ in real projective $(n-1)$-space $\mathbf P^{n-1}(\mathbf R)$. Since $f\_i$ is of odd degree, each vanishing set $Z(f\_i)$ is homologically nontrivial, i.e., represents the nonzero homology class of $H\_{n-2}(\mathbf P^{n-1}(\mathbf R),\mathbf Z/2)$. Since $r\leq n-1$, the intersection product of the vanishing sets is homologically nontrivial. In particular, the intersection $\bigcap\_{i=1}^r Z(f\_i)$ is nonempty, i.e., there is an $a\in\mathbf R^n\setminus\{0\}$ such that $f\_i(a)=0$ for all $i$. This proves the statement above.


By your argument for the case $r=2$, you can then deduce that there is an $a\in\mathbf R^n\setminus\{0\}$ such that $f\_i(a)\geq0$ for all $i$, if $r\leq n$. 


