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Statement

Use the rooted graph, positive integers $a\le b$, and degree $d$ of [[extremal_graph_theory/adamczewski_2026_erdos571/generic_rooted_fibers|the generic-fiber lemma]]. For any extension of fields $K_0\subseteq K$, there are an integer $t\ge1$ and a nonzero polynomial $Q$ over $K_0$ in the coefficients of the $a$ degree-at-most-$d$ polynomials such that

$$Q(c)\ne0\quad\Longrightarrow\quad \text{the polynomial graph with coefficients }c\in K^{aM} \text{ contains no }F^{(t)}.$$

Here $M$ is the number of degree-at-most-$d$ monomials in $2b$ variables. The conclusion is over the fixed field $K$. In the finite-field application $K$ will be an algebraic closure of $K_0$.

Proof

First suppose that no pair consisting of a power and a finite list of nonzero coefficient polynomials has the required property. Index choices by pairs $(m,\mathcal P)$, where $m$ is a positive integer and $\mathcal P$ is a finite set of nonzero polynomials over $K_0$. The supposition gives a coefficient tuple $c_{m,\mathcal P}$ over $K$ which avoids the zeros of every polynomial in $\mathcal P$, yet whose graph contains $F^{(m)}$.

Order these indices by increasing $m$ and inclusion of $\mathcal P$. The sets lying above any fixed index form a proper filter: finitely many such requirements have a common upper bound. Extend it to an ultrafilter and take the field quotient of the corresponding copies of $K$. Let $c_*$ be the tuple of coefficient classes in the quotient field $L$. For every nonzero polynomial $P$ over $K_0$, the indices whose lists contain $P$ form an ultrafilter-large set. Hence $P(c_*)\ne0$. Thus all coordinates of $c_*$ are algebraically independent over $K_0$.

The [[extremal_graph_theory/adamczewski_2026_erdos571/generic_rooted_fibers|generic-fiber lemma]] now supplies $t\ge1$ such that the graph over $L$ defined by $c_*$ contains no $F^{(t)}$. On the other hand, for all indices with $m\ge t$, the chosen copy of $F^{(m)}$ restricts to a copy of $F^{(t)}$. These indices form an ultrafilter-large set.

Those copies pass to a copy in $L$. To spell this out, there are only finitely many assignments of their finitely many vertices to the two sides, so one assignment holds on an ultrafilter-large set. Take the coordinate classes of the chosen vertex images there. Their edge equations pass to the quotient. Injectivity passes as the finite nonvanishing clauses for same-side pairs, while opposite-side pairs are distinct by their side. This is precisely the transfer proved in [[extremal_graph_theory/adamczewski_2026_erdos571/polynomial_compactness|polynomial compactness]]. The resulting copy contradicts the choice of $t$.

A finite list $\mathcal P$ and a power therefore exist. Let $Q=\prod_{P\in\mathcal P}P$, with empty product one. The polynomial ring over a field is an integral domain, so $Q\ne0$. If $Q(c)\ne0$, every factor is nonzero, and the asserted exclusion follows. If the initial argument allowed power zero, replace it by its successor; avoidance of a smaller rooted power implies avoidance of every larger one. Thus we may always take $t\ge1$.

Source and dependencies

The exposition, Proposition 2.1, p. 2, mentions an obstruction polynomial without proving its existence. The pinned formal source supplies PolynomialGerm.copy_germ, lines 3536–3611, and GenericObstruction.finite_obstruction and single_obstruction, lines 3627–3702. The proof above expands that compactness argument and its graph-copy transfer. It does not assume that an arbitrary specialization keeps every individual rooted fiber uniformly bounded; the consequence needed here is exclusion of one rooted power.

Bears on. #571.