Consider system of $8$ equations $$ \alpha^j(1-\alpha)^ip+(1-\alpha)^j \alpha^i (1-p)=q_{j,i} \hspace{1cm} \forall j\in \{0,1,...,7\}, i\in \{0,1,...,7\} \text{ s.t. } i+j=7 $$ where:
- $\{\alpha,p\}$ are the unknowns
- $q_{j,i}$ is known and in $[0,1]$ $\forall j\in \{0,1,...,7\}, i\in \{0,1,...,7\} \text{ s.t. } i+j=7$
- $\alpha\in (\frac{1}{2},1]$, $p\in [0,1]$
Suppose that all the conditions required for the system to have at least one solution wrto $\alpha,p$ are satisfied. Could you help to characterise the set of solutions of the system? Is it singleton?
The answer below is very helpful. However, I've decided to start a bounty because I'm looking for more details on the Grobner basis method (I'm a beginner): why do I need it here? In rough and simple words, what does it consist of? How do we practically implement it for my specific case? What does it give us?
You have $8$ equations in only $2$ unknowns, so having solutions is rather special.
Subtracting the equation for $j=3,i=4$ from the equation for $j=4$,$i=3$ you get $$\alpha^3 (1-\alpha)^3 (2 \alpha - 1) (2 p-1) = 0$$ Now it's easy to check that neither $\alpha = 0$ nor $\alpha = 1$ will work, while if $p = 1/2$ you get (after eliminating $q$) a set of polynomials in $\alpha$ whose greatest common divisor is $2\alpha - 1$. Thus the only way to have a solution is $\alpha = 1/2$. With $\alpha = 1/2$, you must have $q = 1/128$, and $p$ is arbitrary.
But you don't seem to allow $\alpha =1/2$, in which case you are out of luck: there are no other solutions.
EDIT: Replacing $q$ by $q_{j,i}$ (let me call it $q_j$, since $i+j=7$) makes a big difference. Obviously any $p$ and $\alpha$ are possible: just substitute in the equations to find the corresponding values of $q_{j}$. There are equations in the $q_j$ that need to be satisfied: using a Groebner basis in Maple, I find $$ {q_{{5}}}^{6}+7\,{q_{{5}}}^{5}q_{{6}}+5\,{q_{{5}}}^{5}q_{{7}}+16\,{q_{ {5}}}^{4}{q_{{6}}}^{2}+28\,{q_{{5}}}^{4}q_{{6}}q_{{7}}+10\,{q_{{5}}}^{ 4}{q_{{7}}}^{2}+7\,{q_{{5}}}^{3}{q_{{6}}}^{3}+47\,{q_{{5}}}^{3}q_{{7}} {q_{{6}}}^{2}+42\,{q_{{5}}}^{3}q_{{6}}{q_{{7}}}^{2}+10\,{q_{{5}}}^{3}{ q_{{7}}}^{3}-22\,{q_{{5}}}^{2}{q_{{6}}}^{4}+7\,{q_{{5}}}^{2}{q_{{6}}}^ {3}q_{{7}}+45\,{q_{{5}}}^{2}{q_{{7}}}^{2}{q_{{6}}}^{2}+28\,{q_{{5}}}^{ 2}q_{{6}}{q_{{7}}}^{3}+5\,{q_{{5}}}^{2}{q_{{7}}}^{4}-28\,q_{{5}}{q_{{6 }}}^{5}-40\,q_{{5}}{q_{{6}}}^{4}q_{{7}}-7\,q_{{5}}{q_{{6}}}^{3}{q_{{7} }}^{2}+13\,q_{{5}}{q_{{7}}}^{3}{q_{{6}}}^{2}+7\,q_{{5}}q_{{6}}{q_{{7}} }^{4}+q_{{5}}{q_{{7}}}^{5}-8\,{q_{{6}}}^{6}-20\,{q_{{6}}}^{5}q_{{7}}- 18\,{q_{{6}}}^{4}{q_{{7}}}^{2}-7\,{q_{{6}}}^{3}{q_{{7}}}^{3}-{q_{{6}}} ^{2}{q_{{7}}}^{4}-{q_{{6}}}^{5}=0 $$ which describes a certain surface in $q_5, q_6, q_7$ space. $q_4$, $q_3$, $q_2$, $q_1$, $q_0$ are then determined by the values of $q_5, q_6, q_7$. The equation for $\alpha$ is a quadratic: $$\alpha^2-\alpha+q_1+5 q_2+10 q_3+10 q_4+5 q_5+q_6=0$$ so there may be two values of $\alpha$. However, since $\alpha^2 - \alpha$ is strictly increasing for $\alpha \ge 1/2$, only one can be $\ge 1/2$. And finally, the equation for $p$ is linear in $p$, of the form $$ (448 q_6-320 q_7-1) p + f(\alpha, q_1, \ldots, q_7)=0$$ so at least if $448 q_6 - 320 q_7 \ne 1$, $p$ is uniquely determined.
EDIT: Explicitly, this last equation is $$ \left( 448\,q_{{6}}-320\,q_{{7}}-1 \right) p+1+15616\,\alpha\,q_{{5}} q_{{6}}+32768\,\alpha\,q_{{5}}q_{{7}}+q_{{1}}+8\,q_{{2}}+29\,q_{{3}}+ 64\,q_{{4}}+99\,q_{{5}}-136\,q_{{6}}+319\,q_{{7}}-\alpha-58\,\alpha\,q _{{3}}-128\,\alpha\,q_{{4}}-13184\,q_{{4}}q_{{5}}-1664\,q_{{3}}q_{{4}} -16\,\alpha\,q_{{2}}-2\,\alpha\,q_{{1}}-256\,q_{{2}}q_{{7}}-2048\,q_{{ 3}}q_{{7}}-7424\,q_{{4}}q_{{7}}-7808\,q_{{5}}q_{{6}}-16384\,q_{{5}}q_{ {7}}-2432\,q_{{6}}q_{{7}}-15872\,{q_{{5}}}^{2}-6272\,{q_{{4}}}^{2}-192 \,{q_{{3}}}^{2}+26368\,\alpha\,q_{{4}}q_{{5}}+3328\,\alpha\,q_{{3}}q_{ {4}}+12544\,\alpha\,{q_{{4}}}^{2}+512\,\alpha\,q_{{2}}q_{{7}}+4096\, \alpha\,q_{{3}}q_{{7}}+14848\,\alpha\,q_{{4}}q_{{7}}+4864\,\alpha\,q_{ {6}}q_{{7}}+31744\,\alpha\,{q_{{5}}}^{2}+384\,\alpha\,{q_{{3}}}^{2}- 318\,\alpha\,q_{{7}}-16640\,\alpha\,{q_{{6}}}^{2}+640\,\alpha\,{q_{{7} }}^{2}-198\,\alpha\,q_{{5}}-176\,\alpha\,q_{{6}}+8320\,{q_{{6}}}^{2}- 320\,{q_{{7}}}^{2} =0$$