This is in fact related to the previous question that I have asked but I decided that it would be better if I post this as a new question
1)What is a generic matrix(the definitions in net dont really make sense ,I need a simple one with an example)
Also how does it satisfy the property given below?
Notations used:
(i)$C_k:= \{a = (a1, ··· , ak) |1 ≤a1< ··· <ak≤n\}$ denotes the collection of all ordered k-tuples from $\{1, ··· , n\}. $
(ii) Given $ a = (a_1, ..., a_k)∈C_k$;
$X^a= [a_1, ··· , a_k|1, 2, ..., k]$denotes the $k ×k $ minor of the matrix $X$, with $a_1, ..., a_k$ as rows and $1, ..., k$ as columns.
•$S_k:= \{X^a:a ∈C_k\}$and $I_k$ denotes the ideal generated by $S_k$ in the polynomial ring R
The Groebner Basis of the ideal $I_k$ with respect to the monomial ordering is the set $S_k$.
The proof uses Buchberger's criterion
$S(X^c,X^d) \to_{(S_k)} r$. Then the set $S(X^c,X^d) - \sum_{a_i \in C_i}(h_i.X^{a_i}) = r$.
If $X$ is generic, then the set of all $k \times k$ monomial ordering forms a Groebner basis with respect to the the chosen monomial ordering.(strumfels 1990)(**)
Therefore there exists $[a_1,a_2,...,a_k \mid b_1 \cdots b_k]$ such that $\prod_{i=1}^k(\chi_{a_i.b_i})$ divides the $LT(r)$. We see that if $b_k=k$ then we are done.
Let $X$ be generic symmetric .Then, $a_k=k$ and $b_k \ge k+1$ imply that the minor belongs to the set $S_k$.(Can someone explain this property of generic symmetric matrix?)
Let $X$ be generic. Then, for any $a_k$ and under the condition $b_k \ge k+1$, then $\chi_{a_k.b_k} | LT(r)$ but $\chi_{a_k.b_k}$ doesn't divide any term of the elements in $S_k$. (Also I need some help in this property of generic symmetric matrix?)
Also if $X$ is a generic matrix then is there a relation between a generic matrix and a vandermonde matrix ?
I am attaching the link of my previous question.
Confusion in grobner basis and ideals generated by matrices.