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Consider a system $\Sigma (y)$ of $m$ linear equations in $n$ variables $x_1,\cdots,x_n$:

$\sum_{j=1}^n a_{i,j}(y)\cdot x_j=b_i(y)$, $i=1,\cdots,m$,

whose coefficients $a_{i,j}(y)$ and $b_i(y)$ are polynomials over $\mathbb{C}$ in a variable $y$. Suppose for some $y_0\in \mathbb{C}$, $\Sigma (y_0)$ has a unique solution. Then prove:

(1). $n\leq m$;

(2). for all but finitely many $y\in \mathbb{C}$, the system $\Sigma(y)$ has at most one solution;

(3). Furthermore, if $m=n$, then $\Sigma (y)$ has a unique solution for all but finitely many $y\in \mathbb{C}$.

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1 Answer 1

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  1. If a linear system (over a field) has a unique solution, then the number of unknowns has to be no greater than the number of variables.

  2. Having a unique solution means that the rank of the system at $y_0$ is exactly $n.$ The rank dropping corresponds to the determinant of some $n \times n$ minor becoming zero (that determinant is not the zero polynomial by assumption). Since there are finitely many minors, and each has finitely many zeros...

  3. See 2.

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