Manuscript XVI-O1-ST@5 · XVI-O1-ST-v5.tex · theorem · dd45e0d1

Theorem: Varying-response block system

Finite reduction of the full gauge-family row problem · lines 1716–1736 · proof included or linked

XVI-O1-ST@5#thm:blocklift

Let

M (ζ) = a = 0 \sum^{m} M_{a} ζ^{a}, y (ζ) = b = 0 \sum^{q} y_{b} ζ^{b} .

For a declared degree bound $D$ , existence of $c (ζ) = \sum_{j = 0}^{D} c_{j} ζ^{j}$ is equivalent to

a + j = t 0 \leq a \leq m 0 \leq j \leq D \sum M_{a} c_{j} = y_{t}, 0 \leq t \leq max (q, m + D),

where absent target coefficients are zero.

Proof. Expand the product and equate every coefficient, including coefficients above the target degree.

Displayed formulas in this passage

unlabeled · line 1719
$M (ζ) = a = 0 \sum^{m} M_{a} ζ^{a}, y (ζ) = b = 0 \sum^{q} y_{b} ζ^{b} .$
unlabeled · line 1725
$a + j = t 0 \leq a \leq m 0 \leq j \leq D \sum M_{a} c_{j} = y_{t}, 0 \leq t \leq max (q, m + D),$