Additive gaps and an affine-coordinate audit

Definitions and mathematical statement

Pollack's source question asks what is known about gaps between irreducible polynomials over ${\mathbb{F}}_q$. The supplied research interpretation takes a gap to mean an additive difference between monic irreducibles of the same large degree, rather than consecutive polynomials in an unspecified ordering. Fix a prime power $q$, a nonzero $h\in{\mathbb{F}}_q[T]$ of degree $d$, and an integer $n>d$. Define \[ N_q(n;h)=\#\{f\in{\mathbb{F}}_q[T]:f\text{ monic of degree }n,\ f\text{ and }f+h\text{ irreducible}\}. \] The original question is to describe results or bounds for such gaps and counts; it supplies no specific asymptotic formula or limiting regime. Its final isolated text "$>1$" has no stated antecedent and is not imposed here as a further condition.

For a monic irreducible $P$, set $|P|=q^{\deg P}$ and let $\nu_P(h)$ be the number of distinct residues among $0,-h$ modulo $P$. Thus $\nu_P(h)=1$ if $P\mid h$, and $2$ otherwise. Call $\{0,h\}$ admissible when $\nu_P(h)<|P|$ for every such $P$, and define its singular series by \[ \mathfrak S_q(h)=\prod_P \frac{1-\nu_P(h)/|P|}{(1-1/|P|)^2}. \] The supplied candidate synthesis asserts that this product converges and is positive exactly for admissible pairs, with \[ \{0,h\}\text{ admissible}\quad\Longleftrightarrow\quad \begin{cases} \text{always},&q>2,\\ T(T+1)\mid h,&q=2. \end{cases} \] It reports that failure of admissibility forces $N_q(n;h)=0$ for sufficiently large $n$; admissibility alone is not asserted to prove a prime-pair asymptotic.

For fixed $n>d$, $a\in{\mathbb{F}}_q^*$, and $b\in{\mathbb{F}}_q$, define \[ f^{a,b}(T)=a^{-n}f(aT+b),\qquad h^{a,b}(T)=a^{-n}h(aT+b). \] The reported normalized affine invariance preserves monicity, irreducibility, the count $N_q(n;h)$, admissibility and the singular series, with the count for $h$ equal to that for $h^{a,b}$. If the leading coefficient of $h$ is $c$, those of its transformed gaps are precisely \[ c({\mathbb{F}}_q^*)^{n-d}, \] since each transformation gives coefficient $ca^{d-n}$. In particular the condition $\gcd(n-d,q-1)=1$ makes this power-coset all of ${\mathbb{F}}_q^*$. The report identifies this condition in an unnamed published monomial scaling theorem as a restriction of affine transport, distinct from a local obstruction. The theorem's full statement and reference are not supplied, so no additional prescribed-gap existence theorem is inferred.

Short English statement

Study additive differences between monic irreducible polynomials of equal degree. The supplied report identifies the binary local obstruction and separates it from the leading-coefficient restriction imposed by affine changes of variable; it does not supply a general prime-pair asymptotic.

Sources

• Local UnsolvedMath registry, permanent ID $20000391$, source code catalogue entry; AIM algebraic number theory corpus, aim-algebraic-number-theory-notes.json, notes[42]. Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0.
• AIM workshop Arithmetic statistics over finite fields and function fields, Question 13, attributed to Pollack: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf. The extracted question ends with an unexplained "$>1$"; original remarks and literature fields are empty. The verification field ends after "The exact record in the AIM corpus is:".
• The research summary refers to a published monomial scaling theorem without author, title, date or full statement; no bibliographic details are invented.
• Registry assessment dated August 14, 2026: partially_solved, PARTIAL-PROGRESS; attempt 1 labelled partial result, rigor proved, novelty confidence low. Its coordinate-audited candidate synthesis is a machine-generated research aid, not peer reviewed, and requires independent verification before citation.