Theorem: Constant-response polynomial lift

Let $M$ be a constant finite matrix and $y(\zeta )={{\sum }_{j=0}}^q{y}_j{\zeta }^j$, over $\mathbb{K}$. For a complete left kernel $L$, the following are equivalent:

1. $L{y}_j=0$ for every $j$;

2. a polynomial solution $c(\zeta )={{\sum }_{j=0}}^q{c}_j{\zeta }^j$ exists;

3. the target belongs to $\mathrm{im}M$ at every point of an infinite parameter set in the field.

The same original minor and left kernel certify each coefficient solve. With complete right kernel $Z$, all polynomial solutions differ by $Zt(\zeta )$.

Proof. The first condition solves each $M{c}_j=y_j$. A polynomial solution gives pointwise membership. Conversely, $Ly(\zeta )$ vanishes on infinitely many points, so every entry is the zero polynomial. The kernel assertion is coefficientwise.