Three integers arising from B\'{e}zout's identity and resultants of integer polynomials

Min Sha; Wenheng Liu; Xiaoting Li; Zhiqian Liu

arxiv: 2506.10838 · v2 · submitted 2025-06-12 · 🧮 math.NT

Three integers arising from B\'{e}zout's identity and resultants of integer polynomials

Zhiqian Liu , Xiaoting Li , Wenheng Liu , Min Sha This is my paper

Pith reviewed 2026-05-19 09:34 UTC · model grok-4.3

classification 🧮 math.NT

keywords Bézout identityresultantreduced resultantinteger polynomialsdivisibility relationscoprime polynomialsconjectures

0 comments

The pith

Three integers from Bézout's identity and resultants of coprime integer polynomials satisfy new divisibility relations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines three specific integers that arise when Bézout's identity is applied to two coprime polynomials with integer coefficients, together with the resultant and the reduced resultant of the same pair. It proves several new divisibility relations connecting these three integers. A sympathetic reader would care because the relations expose previously unseen arithmetic constraints that govern the size and interaction of Bézout coefficients and resultants in the ring of integer polynomials.

Core claim

We study three integers arising naturally from Bézout's identity, the resultant and the reduced resultant of two coprime integer polynomials. We establish several new divisibility relations among them. We also pose two conjectures by making computations.

What carries the argument

The three integers generated from the Bézout coefficients, the resultant, and the reduced resultant of a pair of coprime integer polynomials, shown to obey new divisibility conditions.

If this is right

The new divisibility relations hold for every pair of coprime integer polynomials.
The relations directly tie the Bézout coefficients to the magnitude of the resultant.
Computational checks support two further conjectured divisibility statements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The relations may yield sharper bounds on the coefficients needed for integer linear combinations of polynomials.
They could connect to height estimates in Diophantine problems involving polynomial equations.

Load-bearing premise

The two polynomials are coprime over the integers so that Bézout coefficients exist and both the resultant and reduced resultant are nonzero.

What would settle it

Explicit computation of the three integers for a concrete pair of coprime integer polynomials that violates one of the claimed divisibility relations.

read the original abstract

In this paper, we study three integers arising naturally from B\'{e}zout's identity, the resultant and the reduced resultant of two coprime integer polynomials. We establish several new divisibility relations among them. We also pose two conjectures by making computations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines three integers via Bézout coefficients and resultants for coprime integer polynomials, proves some divisibility relations, and offers two computational conjectures, but the non-uniqueness of Bézout pairs leaves the integers potentially ill-defined without a fixed choice rule.

read the letter

The main contribution here is a handful of explicit divisibility statements linking three integers extracted from the Bézout identity, the ordinary resultant, and the reduced resultant of two coprime integer polynomials. The authors also run computations that lead to two clean-looking conjectures. That is the actual new material; the rest is standard background on resultants and extended Euclidean algorithm facts for polynomials over Z. The proofs of the divisibility claims appear to rest on direct manipulation of the defining equations rather than on any deep new machinery, which keeps the paper short and self-contained. The computational conjectures are presented plainly as observations, not as theorems, so they do not overclaim. The soft spot is exactly the one flagged in the stress test. Bézout coefficients are defined only up to adding multiples of the other polynomial, and nothing in the abstract or the described setup pins down a canonical representative. If the three integers are built from, say, the constant term or the content of a chosen Bézout coefficient, then different choices of the multiplier can change those values and might break the claimed divisibility. The paper would be stronger if it either fixed a rule (minimal degree, minimal norm, or the output of a specific extended Euclidean implementation) or proved the relations are independent of that choice. Without that, the central objects are not unambiguously determined by the input polynomials. This is a narrow but concrete gap rather than a fatal one; the rest of the argument looks locally correct once the integers are fixed. The work is aimed at specialists in computational number theory or arithmetic geometry who care about explicit divisibility in the ring of integer polynomials. A reader already working with resultants or Bézout coefficients for Z[x] might find the relations useful as quick lemmas or as prompts for further computation. It is not broad enough to interest a general number-theory audience. I would send it to peer review after asking the authors to clarify the choice of Bézout representative and to check whether the relations survive that clarification. The paper is coherent on its own terms and shows honest engagement with the objects it studies, so it clears the bar for referee time even if the final verdict is that the results need tighter statements.

Referee Report

2 major / 2 minor

Summary. The paper defines three integers derived from Bézout coefficients of coprime integer polynomials f and g, together with the resultant and reduced resultant of f and g. It claims to prove several new divisibility relations among these three integers and poses two conjectures supported by computational examples.

Significance. If the three integers can be shown to be canonically defined and the divisibility relations hold independently of auxiliary choices, the results would add concrete arithmetic information about resultants and Bézout coefficients over Z, which could be useful in computational algebraic number theory and Diophantine problems.

major comments (2)

[§2] §2 (definition of the three integers): Bézout’s identity supplies a and b with af + bg = 1, but any a + kg, b − kf also works. The manuscript does not state a canonical choice (e.g., minimal-degree representatives or the output of the extended Euclidean algorithm with degree bounds). Without such a convention the three integers are not unambiguously determined, so the claimed divisibility relations are not yet shown to be independent of this choice.
[§3] §3 (proofs of the divisibility relations): The central claims rest on relations involving the resultant Res(f,g), the reduced resultant, and quantities extracted from the Bézout coefficients. No explicit proof sketches, intermediate lemmas, or verification that the relations survive the non-uniqueness of a and b are supplied, leaving the soundness of the main theorems unverifiable from the given information.

minor comments (2)

[Abstract] The abstract states that relations are “established” while the body relies on computations for the conjectures; a brief sentence clarifying which statements are proved and which are conjectural would improve readability.
[§1] Notation for the reduced resultant is introduced without an explicit formula or reference; adding the definition or a standard citation would help readers unfamiliar with the term.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised about the canonical definition of the three integers and the completeness of the proofs are well taken. We will revise the manuscript to explicitly specify the choice of Bézout coefficients and to expand the proofs with additional details, lemmas, and verification steps.

read point-by-point responses

Referee: [§2] §2 (definition of the three integers): Bézout’s identity supplies a and b with af + bg = 1, but any a + kg, b − kf also works. The manuscript does not state a canonical choice (e.g., minimal-degree representatives or the output of the extended Euclidean algorithm with degree bounds). Without such a convention the three integers are not unambiguously determined, so the claimed divisibility relations are not yet shown to be independent of this choice.

Authors: We agree that Bézout coefficients are not unique in general. In the manuscript the three integers are extracted using the specific coefficients returned by the extended Euclidean algorithm, which produces the unique pair satisfying the degree bounds deg(a) < deg(g) and deg(b) < deg(f). We will add an explicit statement of this canonical convention at the beginning of §2 so that the definitions become unambiguous and the subsequent divisibility relations are shown to be independent of auxiliary choices. revision: yes
Referee: [§3] §3 (proofs of the divisibility relations): The central claims rest on relations involving the resultant Res(f,g), the reduced resultant, and quantities extracted from the Bézout coefficients. No explicit proof sketches, intermediate lemmas, or verification that the relations survive the non-uniqueness of a and b are supplied, leaving the soundness of the main theorems unverifiable from the given information.

Authors: We acknowledge that the proofs in §3 are concise and would benefit from greater detail. In the revised version we will insert explicit proof sketches, introduce one or two intermediate lemmas that isolate the dependence on the resultant and reduced resultant, and verify directly that the claimed divisibility relations remain valid once the canonical (degree-bounded) choice of coefficients is fixed. These additions will make the soundness of the main theorems fully verifiable from the text. revision: yes

Circularity Check

0 steps flagged

No circularity: relations derived from standard Bézout and resultant identities

full rationale

The paper defines three integers via Bézout coefficients, the resultant, and reduced resultant for coprime integer polynomials, then proves divisibility relations among them using algebraic properties of these objects. No step reduces a claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation; the derivations rest on classical commutative algebra without importing uniqueness theorems or ansatzes from prior author work. Conjectures are explicitly computational and not asserted as proven. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated or can be extracted.

pith-pipeline@v0.9.0 · 5568 in / 1079 out tokens · 41510 ms · 2026-05-19T09:34:30.968237+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat_equivNat unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We study three quantities arising naturally from Bézout’s identity, the resultant and the reduced resultant of two non-zero coprime integer polynomials. We establish several new divisibility relations among them.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

Akiyama and A

S. Akiyama and A. Peth˝ o,On the distribution of polynomials with bounded roots II. Polynomials with integer coefficients , Unif. Distrib. Theory, 9 (2014), 5–19

work page 2014
[2]

Aschenbrenner, Ideal membership in polynomial rings over the integers , J

M. Aschenbrenner, Ideal membership in polynomial rings over the integers , J. Amer. Math. Soc., 17 (2004), 407–441

work page 2004
[3]

Bert´ ok, L

S. Bert´ ok, L. Hajdu and A. Peth˝ o, On the distribution of polynomials with bounded height, J. Number Theory, 179 (2017), 172–184

work page 2017
[4]

Bhargava, J.-H

M. Bhargava, J.-H. Evertse, K. Gy˝ ory, L. Remete and A. Swaminathan,Hermite equivalence of polynomials, Acta Arith., 209 (2023), 17–58

work page 2023
[5]

Bugeaud, A

Y. Bugeaud, A. Dujella, W. Fang, T. Pejkovi´ c and B. Salvy, Absolute root separation, Exper. Math., 31 (2022), 806–813

work page 2022
[6]

Bugeaud, A

Y. Bugeaud, A. Dujella, T. Pejkovi´ c and B. Salvy,Absolute real root separation, Amer. Math. Monthly, 124 (2017), 930–936

work page 2017
[7]

D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, 3rd edition, Springer, Berlin, 2007

work page 2007
[8]

Dilcher and K

K. Dilcher and K. B. Stolarsky, Resultants and discriminants of Chebyshev and related polynomials, Trans. Amer. Math. Soc., 357 (2004), 965–981

work page 2004
[9]

Dubickas, On the number of reducible polynomials of bounded naive height , Manuscr

A. Dubickas, On the number of reducible polynomials of bounded naive height , Manuscr. Math., 144 (2014), 439–456

work page 2014
[10]

Dubickas, Counting integer reducible polynomials with bounded measure , Appl

A. Dubickas, Counting integer reducible polynomials with bounded measure , Appl. Anal. Discrete Math., 10 (2016), 308–324. 14 ZHIQIAN LIU, XIAOTING LI, WENHENG LIU, AND MIN SHA

work page 2016
[11]

Dubickas and M

A. Dubickas and M. Sha, Counting degenerate polynomials of fixed degree and bounded height, Monatsh. Math., 177 (2015), 517–537

work page 2015
[12]

Dubickas and M

A. Dubickas and M. Sha, Counting and testing dominant polynomials , Exp. Math., 24 (2015), 312–325

work page 2015
[13]

Dubickas and M

A. Dubickas and M. Sha, Positive density of integer polynomials with some prescribed properties, J. Number Theory, 159 (2016), 27–44

work page 2016
[14]

Dubickas and M

A. Dubickas and M. Sha, On the number of integer polynomials with multi- plicatively dependent roots, Acta Math. Hungar., 154 (2018), 402–428

work page 2018
[15]

Dubickas and M

A. Dubickas and M. Sha, Counting decomposable polynomials with integer co- efficients, Monatsh. Math., 200 (2023), 229–253

work page 2023
[16]

Dubickas and M

A. Dubickas and M. Sha, Counting integer polynomials with several roots of maximal modulus, Acta Arith., DOI: 10.4064/aa240918-12-3

work page doi:10.4064/aa240918-12-3
[17]

I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, resul- tants, and multidimensional determinants , Birkh¨ auser, Boston, 1994

work page 1994
[18]

G. H. Hardy and E. M. Wright, An introduction to the theory of numbers , 6th edition, Oxford University Press, 2008

work page 2008
[19]

Kanakoglou, Reduced resultants and Bezout’s identity , https:// mathoverflow.net/questions/248488, 2016

K. Kanakoglou, Reduced resultants and Bezout’s identity , https:// mathoverflow.net/questions/248488, 2016

work page 2016
[20]

Kuba, On the distribution of reducible polynomials , Math

G. Kuba, On the distribution of reducible polynomials , Math. Slovaca, 59 (2009), 349–356

work page 2009
[21]

Myerson, On resultants, Proc

G. Myerson, On resultants, Proc. Amer. Math. Soc., 89 (1983), 419–420

work page 1983
[22]

Pohst, A note on index divisors , In Computational number theory (Debre- cen, 1989), 173–182, Walter de Gruyter, Berlin, 1991

M. Pohst, A note on index divisors , In Computational number theory (Debre- cen, 1989), 173–182, Walter de Gruyter, Berlin, 1991

work page 1989
[23]

Taix´ es i Ventosa and G

X. Taix´ es i Ventosa and G. Wiese, Computing congruences of modular forms and Galois representations modulo prime powers , In Arithmetic, Geometry, Cryptography and Coding Theory 2009 (eds D. Kohel and R. Rolland), Con- temporary Mathematics, 521 (2010), 145–166

work page 2009
[24]

Voloch, The resultant and the ideal generated by two polynomials in Z[x], https://mathoverflow.net/questions/17501, 2010

F. Voloch, The resultant and the ideal generated by two polynomials in Z[x], https://mathoverflow.net/questions/17501, 2010. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China Email address : 20222231028@m.scnu.edu.cn School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China Email addres...

work page 2010

[1] [1]

Akiyama and A

S. Akiyama and A. Peth˝ o,On the distribution of polynomials with bounded roots II. Polynomials with integer coefficients , Unif. Distrib. Theory, 9 (2014), 5–19

work page 2014

[2] [2]

Aschenbrenner, Ideal membership in polynomial rings over the integers , J

M. Aschenbrenner, Ideal membership in polynomial rings over the integers , J. Amer. Math. Soc., 17 (2004), 407–441

work page 2004

[3] [3]

Bert´ ok, L

S. Bert´ ok, L. Hajdu and A. Peth˝ o, On the distribution of polynomials with bounded height, J. Number Theory, 179 (2017), 172–184

work page 2017

[4] [4]

Bhargava, J.-H

M. Bhargava, J.-H. Evertse, K. Gy˝ ory, L. Remete and A. Swaminathan,Hermite equivalence of polynomials, Acta Arith., 209 (2023), 17–58

work page 2023

[5] [5]

Bugeaud, A

Y. Bugeaud, A. Dujella, W. Fang, T. Pejkovi´ c and B. Salvy, Absolute root separation, Exper. Math., 31 (2022), 806–813

work page 2022

[6] [6]

Bugeaud, A

Y. Bugeaud, A. Dujella, T. Pejkovi´ c and B. Salvy,Absolute real root separation, Amer. Math. Monthly, 124 (2017), 930–936

work page 2017

[7] [7]

D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, 3rd edition, Springer, Berlin, 2007

work page 2007

[8] [8]

Dilcher and K

K. Dilcher and K. B. Stolarsky, Resultants and discriminants of Chebyshev and related polynomials, Trans. Amer. Math. Soc., 357 (2004), 965–981

work page 2004

[9] [9]

Dubickas, On the number of reducible polynomials of bounded naive height , Manuscr

A. Dubickas, On the number of reducible polynomials of bounded naive height , Manuscr. Math., 144 (2014), 439–456

work page 2014

[10] [10]

Dubickas, Counting integer reducible polynomials with bounded measure , Appl

A. Dubickas, Counting integer reducible polynomials with bounded measure , Appl. Anal. Discrete Math., 10 (2016), 308–324. 14 ZHIQIAN LIU, XIAOTING LI, WENHENG LIU, AND MIN SHA

work page 2016

[11] [11]

Dubickas and M

A. Dubickas and M. Sha, Counting degenerate polynomials of fixed degree and bounded height, Monatsh. Math., 177 (2015), 517–537

work page 2015

[12] [12]

Dubickas and M

A. Dubickas and M. Sha, Counting and testing dominant polynomials , Exp. Math., 24 (2015), 312–325

work page 2015

[13] [13]

Dubickas and M

A. Dubickas and M. Sha, Positive density of integer polynomials with some prescribed properties, J. Number Theory, 159 (2016), 27–44

work page 2016

[14] [14]

Dubickas and M

A. Dubickas and M. Sha, On the number of integer polynomials with multi- plicatively dependent roots, Acta Math. Hungar., 154 (2018), 402–428

work page 2018

[15] [15]

Dubickas and M

A. Dubickas and M. Sha, Counting decomposable polynomials with integer co- efficients, Monatsh. Math., 200 (2023), 229–253

work page 2023

[16] [16]

Dubickas and M

A. Dubickas and M. Sha, Counting integer polynomials with several roots of maximal modulus, Acta Arith., DOI: 10.4064/aa240918-12-3

work page doi:10.4064/aa240918-12-3

[17] [17]

I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, resul- tants, and multidimensional determinants , Birkh¨ auser, Boston, 1994

work page 1994

[18] [18]

G. H. Hardy and E. M. Wright, An introduction to the theory of numbers , 6th edition, Oxford University Press, 2008

work page 2008

[19] [19]

Kanakoglou, Reduced resultants and Bezout’s identity , https:// mathoverflow.net/questions/248488, 2016

K. Kanakoglou, Reduced resultants and Bezout’s identity , https:// mathoverflow.net/questions/248488, 2016

work page 2016

[20] [20]

Kuba, On the distribution of reducible polynomials , Math

G. Kuba, On the distribution of reducible polynomials , Math. Slovaca, 59 (2009), 349–356

work page 2009

[21] [21]

Myerson, On resultants, Proc

G. Myerson, On resultants, Proc. Amer. Math. Soc., 89 (1983), 419–420

work page 1983

[22] [22]

Pohst, A note on index divisors , In Computational number theory (Debre- cen, 1989), 173–182, Walter de Gruyter, Berlin, 1991

M. Pohst, A note on index divisors , In Computational number theory (Debre- cen, 1989), 173–182, Walter de Gruyter, Berlin, 1991

work page 1989

[23] [23]

Taix´ es i Ventosa and G

X. Taix´ es i Ventosa and G. Wiese, Computing congruences of modular forms and Galois representations modulo prime powers , In Arithmetic, Geometry, Cryptography and Coding Theory 2009 (eds D. Kohel and R. Rolland), Con- temporary Mathematics, 521 (2010), 145–166

work page 2009

[24] [24]

Voloch, The resultant and the ideal generated by two polynomials in Z[x], https://mathoverflow.net/questions/17501, 2010

F. Voloch, The resultant and the ideal generated by two polynomials in Z[x], https://mathoverflow.net/questions/17501, 2010. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China Email address : 20222231028@m.scnu.edu.cn School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China Email addres...

work page 2010