Exact polynomial calculus degree p for the onto functional pigeonhole principle with surplus divisible by p
Year: 2026
Type: preprint
Abstract: ## SettingThe onto functional pigeonhole principle $\mathrm{onto\text{-}FPHP}^{n+k}_n$ says that $n+k$pigeons can be mapped one-to-one onto $n$ holes. It is false for every $k\ge1$. As a system ofpolynomial equations in Boolean variables $x_{ij}$ (pigeon $i$ sits in hole $j$) it has theaxioms $\sum_j x_{ij}-1$ (each pigeon sits in a hole), $\sum_i x_{ij}-1$ (each hole holds exactlyone pigeon), $x_{ij}x_{i'j}$ (no two pigeons share a hole) and $x_{ij}x_{ij'}$ (no pigeon sits intwo holes). This is the linear encoding used in the literature on this family[1, 2]. These results are for the linear encoding only: translating the CNF clause by clause instead turns each pigeon clause into a polynomial of degree $n$ [2], so degree bounds below $n$ say nothing about that encoding.A polynomial calculus (PC) refutation over the field $\mathbb F_p$ derives the polynomial $1$from the axioms and the Boolean axioms $x^2-x$, using linear combinations and multiplication by avariable. Its degree is the largest degree of a line. The PC degree of the principle is the leastdegree of a refutation. When $p$ does not divide $k$, summing the pigeon axioms and subtractingthe hole axioms gives the constant $-k$, a degree-1 refutation. The question is what happens when$p$ divides $k$.## ResultFor every prime $p$, every $c$ not divisible by $p$, and every $n\ge 2p-3$, the PC degree of$\mathrm{onto\text{-}FPHP}^{n+pc}_n$ over $\mathbb F_p$ is exactly $p$: there is a refutation ofdegree $p$, and none of degree $p-1$. More generally, whenever $p$ divides $k$ and $n\ge2p-3$,there is no refutation of degree $p-1$. The same formula therefore has PC degree $p$ over$\mathbb F_p$ and PC degree $1$ over $\mathbb F_q$ for every prime $q$ not dividing $pc$.## Why it mattersRazborov's survey asks (Question 7) whether $\mathrm{onto\text{-}FPHP}^{n+p^\ell}_n$ hasbounded-degree PC refutations in characteristic $p$ [2]. The upper bound $p$ atthe first level comes from Riis's criterion [3]. The known lower bounds arefor the weaker Nullstellensatz system [1] and, for PC, a bound that grows with$\ell$ for large $n$ [4]. This result fixes the degree exactly at the firstlevel (surplus divisible by $p$ but not by $p^2$), for every prime, from a small linear thresholdin $n$.## Proved vs checkedEverything is proved in Lean 4 and kernel-checked, for all primes $p$ and all $n\ge2p-3$; nothingis checked only up to a bound. The proofs use only the standard axioms `propext`,`Classical.choice` and `Quot.sound`. A second agent in the run, working separately from theauthor, re-proved each of the three statements in Lean. No person has reviewed the proofs; theevidence is the kernel check and the independent re-proof.The Lean statements are for the library's formulation of PC, in which lines are functions onthe Boolean cube. A later theorem of the same library shows that a refutation of degree $D$ instandard (syntactic) PC over $\mathbb F_p$ exists exactly when one exists in the library'sformulation, so the result holds verbatim for standard PC (`std_pc_degree_exact`).This result extends:- Riis-Sitharam, ECCC TR97-048, Theorem 33: PC degree >= 2^l - 1 for onto-PHP^{N+p^l}_N over characteristic p once N >= (21+3p^2)(2^l - 2) - p (same family; proof not checked by us; at l = 2 this is >= 3 against our >= p+1 from n = 2p-1; at p = 2, l = 3 it is >= 7 for N >= 196 against our >= 5 for n >= 7).- Beame-Riis 1998, Theorem 8.1 and Lemma 8.7: Nullstellensatz degree between 2^l - 1 and p^l - 1 (multiplier degree) for the same family; Section 10 lists polynomial-calculus lower bounds as open.- Riis 1993, as Theorem 8 of Razborov's survey: degree-d PC refutations when C(m,d) and C(n,d) differ mod p, which gives the matching upper bound p for k = pc, p not dividing c (via Lucas).- Razborov, Proof complexity of pigeonhole principles (2001), Question 7: the fixed-p, fixed-l bounded-degree question for onto-FPHP^{n+p^l}_n.- Miksa-Nordstrom (JACM 2024): general PC degree method; onto-FPHP with m = n+1 easy (Riis); the case of characteristic dividing the surplus not covered.- BGIP 2001: proves linear PC degree for Tseitin mod p over characteristic q containing a primitive root of unity of order p (Thm 15) and for MOD_p^n over F_q, q not dividing p (Cor 20: "requires degree > delta n"). This is the cross-characteristic regime. Our results concern the same-characteristic regime p ∣ k, so there is no overlap. The paper does not treat onto-PHP.- Alekhnovich-Razborov 2003: "immunity" method giving linear PC degree bounds for mod_p Tseitin over characteristic different from p, for flow tautologies and for random k-CNF in characteristic 2 (abstract only). No PHP or onto-PHP result. The method is relevant only as possible technique.- Impagliazzo-Pudlak-Sgall 1999: simpler proof of Razborov's ceil(n/2)+1 PC lower bound for (injective, non-onto) PHP^m_n over any field, plus an upper bound in large characteristic (abstract). This is a different principle: without the onto and functional axioms the bound is independent of p. It does not transfer to the exact degree p or p+1 for onto-FPHP.- BIKPRS 1996/97: Nullstellensatz lower bounds for Count_q over Z_m with q not dividing m (Thm 3.2) and over Q (Thm 4.6), plus the PC-to-NS simulation (Thm 5.4). This is Nullstellensatz and Count, not PC degree for onto-FPHP, and the paper excludes the case q ∣ m.- Grigoriev FOCS 1998 (snippet only, not opened): Nullstellensatz degree for Tseitin mod p over other characteristics. It is a predecessor of BGIP; NS, not PC, and a different principle.- Filmus-Lauria-Mikša-Nordström-Vinyals 2025: PC space and separations. It mentions the functional PHP only for space limits, not for characteristic-dependent degree.- Ken 2022/2024: restates Razborov's injPHP bound "no PC-refutation ... with degree <= m/2" (Thm 24) and extends it to NS over finite rings (Cor 25). Onto-PHP appears only in reductions (Props 12, 15), and no degree bound for it is proved.- Reduction transfer: none of the opened sources gives a low-degree reduction from Count_q, Tseitin mod q or Mod_q (q != p) to onto-FPHP^{n+k}_n with p ∣ k over F_p. Since the cross-characteristic lower bounds need q != p, they do not yield results 1-3 directly.Our prior-art search (2026-11-06) covered:- Riis, Sitharam, Non-constant degree lower bounds imply linear degree lower bounds, ECCC TR97-048 (1997), pp. 36-38 and 45-46 (Theorem 33, Propositions 27 and 28), opened 2026-10-29- Beame, Riis, More on the relative strength of counting principles, DIMACS Series 39 (1998), homes.cs.washington.edu/~beame/papers/riis.pdf: Theorem 8.1, Lemma 8.7, Section 10, opened 2026-10-27 and 2026-11-01- Razborov, Proof complexity of pigeonhole principles, DLT 2001 (LNCS 2295), people.cs.uchicago.edu/~razborov/files/php_survey.pdf: Theorem 8, Question 7, footnote 5, opened 2026-10-27 and 2026-10-29- Riis, BRICS Report RS-99-29, opened 2026-10-29- Miksa, Nordstrom, A generalized method for proving polynomial calculus degree lower bounds (CCC 2015; JACM 2024; arXiv 1505.01358), opened 2026-10-27- de Rezende et al., Theory of Computing 21(6) (2025), opened 2026-10-29- Lauria et al., Theory of Computing 21(4) (2025), opened 2026-10-27- Buss, Grigoriev, Impagliazzo, Pitassi, "Linear gaps between degrees for the polynomial calculus modulo distinct primes", JCSS 62(2):267-289, 2001 (STOC 1999). Opened: journal final PDF, https://mathweb.ucsd.edu/~sbuss/ResearchWeb/lineargaps/journalfinal.pdf- Alekhnovich, Razborov, "Lower bounds for polynomial calculus: non-binomial case", Proc. Steklov Inst. Math. 242 (2003) 18-35 (FOCS 2001). Opened: abstract only, https://mathnet.ru/eng/tm403 (full text not opened).- Impagliazzo, Pudlak, Sgall, "Lower bounds for the polynomial calculus and the Groebner basis algorithm", Computational Complexity 8(2):127-144, 1999 (ECCC TR97-042). Opened: ECCC abstract only, https://eccc.weizmann.ac.il/report/1997/042/- Buss, Impagliazzo, Krajicek, Pudlak, Razborov, Sgall, "Proof complexity in algebraic systems and bounded depth Frege systems with modular counting", Computational Complexity 6(3):256-298, 1996/97. Opened: full PDF, https://mathweb.ucsd.edu/~sbuss/ResearchWeb/nullstellensatz/token.pdf- Grigoriev, "Tseitin's tautologies and lower bounds for Nullstellensatz proofs", FOCS 1998, pp. 648-652. Snippet only, not opened (dblp/researchr entries).- Mikša, Nordström, "A generalized method for proving polynomial calculus degree lower bounds", CCC 2015 (arXiv 1505.01358). Already credited. Seen in search results; not re-opened here.- Filmus, Lauria, Mikša, Nordström, Vinyals, "Towards an understanding of polynomial calculus: new separations and lower bounds", Theory of Computing 21(4), 2025 (ICALP 2013). Opened: abstract page, https://zesty2.cs.uchicago.edu/articles/v021a004- Eitetsu Ken, "On some Sigma_0^B-formulae generalizing counting principles over V^0", arXiv 2203.10237 (2022; a journal version appears to be Arch. Math. Logic 2024, https://link.springer.com/article/10.1007/s00153-024-00938-1). Opened: ar5iv HTML and Springer page, first ~100k characters of each.- Web searches for 2000-2026 papers on PC degree of onto-PHP or Count where the characteristic divides the surplus: no further hits beyond the items above.Prior-art answer recorded by the owner (basis stated in the file); published in `prior-art/`.If you know of related work we missed, please open an issue.This result was found by AI research agents (the autoresearch harness). It was proved in Lean; the verification steps below build the published programme library.The declaring Lean sources are in `checks/library/`.
Author: Lindheim, Björn
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225425651