Goldbach theorems for group semidomains

Advaith Mopuri; Charles Zhang; Eddy Li

REVIEW

Goldbach theorems for group semidomains

Eddy Li, Advaith Mopuri, Charles Zhang

Commutative Algebra · Submitted 30 Nov 2024 · 16 references

Pith's verdict: The polynomial Goldbach theorem is a solid extension of Kaplan–Polo; the series theorem is not proved as written.

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arXiv ID2412.00590 (version 2)

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CitationAdvaith Mopuri, Charles Zhang, Eddy Li. "Goldbach theorems for group semidomains." arXiv:2412.00590 (2024). https://pith.science/paper/GALGP32W

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Editor's note

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The real contribution here is Theorem 3.6: for additively reduced, additively Furstenberg semidomains S and torsion-free abelian G, the Goldbach property for S[G] is equivalent to A+(S)=S×. That is a genuine extension of Kaplan–Polo's theorem, and Example 3.7 shows the new hypothesis is non-vacuous. The monolithicity lemmas (3.2, 3.3) are clean, and the overall polynomial proof is recoverable. One localized gap: in Case 1 of the proof, the expression f = [f' + s1x^{g1}] + [f'' - s1x^{g1}] uses subtraction that does not exist in S[G], and the term is absent when 2g1=g0. This is repairable by a direct rebalancing; I would not sink the paper on it.

The series half is another story. Theorem 4.6 as stated omits the additively Furstenberg hypothesis that the proof and Section 4's introduction rely on, so the statement is internally inconsistent. More seriously, the proof of (1)⇒(2) is a sketch. It asserts the existence of p and q whose gaps grow strictly, and eventually strictly, respectively, partitioning the support of f. No construction is given, and for the dense case G=Z, g_i=i, such a partition cannot exist: a subset of N with strictly increasing gaps has density 0, so its complement has arbitrarily long runs of consecutive integers and therefore has gaps of size 1 infinitely often. That contradicts the required eventual strict increase. The construction is not merely under-specified; it is impossible for the simplest dense support. Whether the theorem is true is open, but the proof as written does not establish it. Corollary 4.7 inherits this.

The paper is honest in attribution and the writing is clear, but the series section needs a serious rewrite, not copy-editing. For the polynomial result, I would trust the argument after a fix; for the series result, I would ask for a complete construction or a restricted statement (e.g., supports with large gaps).

Who is this for? People in semiring factorization will want Theorem 3.6 and Example 3.7. I would cite the polynomial part. The series part, in current form, I would not rely on. It deserves peer review because the main equivalence is meaningful and most of the work is sound, but the referee report should demand a real proof for Theorem 4.6.

Referee report

Major objections (4)

[Section 3, proof of Theorem 3.6, Case 1] The displayed identity f = [f' + s1 x^{g1}] + [f'' - s1 x^{g1}] is not legitimate in a semiring because additive inverses are not available. Moreover, when 2g1 = g0, the coefficient of x^{g1} is not present in f'', so the expression f'' - s1 x^{g1} is undefined. This is a real gap in a case of the main equivalence, though it is local: if one decomposes s1 = u + v with u, v in S×, then f = (f' + u x^{g1}) + (f'' + v x^{g1}) is a valid splitting, and the irreducibility of the two summands should be verified explicitly.
[Section 3, proof of Theorem 3.6, (3) implies (1)] The claim that f = sum_{i=0}^n s x^{g_i} cannot be written as a sum of n irreducibles is not justified. The additive atomicity of the coefficient s shows that each monomial cannot be split into two nonzero summands, but it does not rule out an irreducible summand containing several of these monomials; indeed, under the gap hypotheses of Lemmas 3.2 and 3.3 such multi-monomial expressions can be irreducible. A separate argument is needed to show that an additive atom outside S× obstructs a bounded Goldbach decomposition.
[Section 4, statement of Theorem 4.6] The theorem as stated assumes only that S is an additively reduced semidomain, but the proof of (3) implies (1) uses the additively Furstenberg property to assert that A+(S) is nonempty, and the proof of (1) implies (2) uses it to decompose coefficients as a unit plus an element of S. Thus the statement is not the theorem being proved. The additively Furstenberg hypothesis should be added to the statement, or a proof under the weaker hypothesis should be supplied. Corollary 4.7 inherits the same issue.
[Section 4, proof of Theorem 4.6, Cases 1 and 2] The existence of the series p and q satisfying properties (a)-(e) is asserted but not proved. Phrases such as "one can construct" and "Lemma 4.4 guarantees" do not specify the partition of the tail of supp(f) into supp(p) and supp(q), nor do they verify the gap conditions needed for Lemma 4.2 and Corollary 4.5. In addition, the later modification q'' := q - v2 x^{t_{M+2}} - u3 x^{t_{M+3}} uses subtraction in a semiring, and no argument is given that inserting the terms v2 x^{t_{M+2}} and u3 x^{t_{M+3}} into p preserves the strict gap monotonicity on which the quoted monolithicity lemmas depend. The proof must supply an explicit construction of p and q and a verification of all gap conditions after every modification.

Minor objections (4)

[Section 2, Definition of S[[G]]] The canonical representation of a series is described by "gi < g_{i+1} for every i" and later "if s_i = 0 then s_{i+1} = 0", but the support is then defined as the set of exponents with s_i nonzero; the interaction between these conventions should be stated more explicitly.
[Lemma 4.4] The conclusion says the sequence (n_i)_{i in N0} consists of positive integers, but the construction begins with n0 = 0; the wording should say nonnegative integers.
[Example 3.7] The final assertion that for every s in S_pq^* there exists t_s with (q/(p+q))^{t_s} dividing s in (S_pq, +) is stated as "not hard to show" but no derivation is given; a short verification would make the example self-contained.
[Throughout Section 4] The proof of Theorem 4.6 uses notation p', p'', q, q', q'' in a way that is sometimes ambiguous, especially because q' is introduced as a finite polynomial while q is a series; renaming these objects would improve readability.

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The paper

The paper claims

This paper proves that, for group semidomains $S[G]$, the Goldbach property holds exactly when the additive atoms of $S$ are its multiplicative units.

The reading

A semidomain is a ring-like object without additive inverses that lives inside an integral domain. The paper asks when polynomial expressions built from a coefficient semidomain $S$ and a torsion-free abelian group $G$ obey a Goldbach-type law: every expression with more than one term splits as a sum of at most two irreducibles. The answer it proves is a single equality inside $S$: this happens exactly when every additive atom of $S$ is a unit and every unit is an additive atom, written $\mathscr{A}_+(S)=S^\times$. The paper also proves the series analogue, where three irreducibles suffice, for groups whose bounded-below subsets are well-ordered and for finitely generated torsion-free groups. If true, the result pins the whole Goldbach phenomenon for these semirings on a checkable condition on coefficients rather than on the group.

What carries the argument

The load-bearing object is the equality $\mathscr{A}_+(S)=S^\times$ between additive atoms and multiplicative units of the coefficient semidomain. Around it sit three supporting mechanisms: the notion of a monolithic polynomial (every factorization has a monomial factor), the support-geometry lemmas (Lemmas 3.2 and 3.3) that certify monolithicity from the spacing of exponents, and the additively Furstenberg hypothesis, which lets every coefficient split as $u+v$ with $u$ a unit. Lemma 3.1 then says a monolithic polynomial whose coefficients have gcd $1$ is irreducible, and the main proof assembles the $u$-parts of the top and bottom coefficients into two such pieces. This machinery converts a purely additive property of $S$ into a factorization statement about $S[G]$.

Extended reading notes

Core claim

The central discovery is Theorem 3.6. If $S$ is an additively reduced and additively Furstenberg semidomain and $G$ is a torsion-free abelian group, then every $f\in S[G]$ with $|\operatorname{supp}(f)|>1$ is a sum of at most two irreducibles if and only if $\mathscr{A}_+(S)=S^\times$, and this condition is also equivalent to the weaker-looking bound at most $n$ irreducibles for any fixed $n>1$. Necessity is forced by a multi-term obstruction: a non-unit additive atom $s$ makes $sx^{g_0}+sx^{g_1}+sx^{g_2}$ impossible to write as two irreducibles. Sufficiency is constructive: using the Furstenberg splitting of each coefficient into a unit plus a remainder, the proof redistributes the unit parts into two monolithic polynomials whose top and bottom coefficients are units, which Lemma 3.1 certifies as irreducible. The same mechanism, with three pieces instead of two, yields the series-level analogue in Theorem 4.6, and Corollary 4.7 extends it to all finitely generated torsion-free groups.

Load-bearing premise

The load-bearing premise is that $S$ is additively Furstenberg, meaning every nonzero element of $S$ splits as an additive atom plus another element of $S$; if this splitting is unavailable for some coefficient, the two-piece decomposition has no starting point and the theorem's equivalence is not established.

Editorial extensions

If this is right

For every additively reduced, additively Furstenberg semidomain $S$ and torsion-free abelian group $G$, the Goldbach property of $S[G]$ is decided entirely by $\mathscr{A}_+(S)=S^\times$; no special structure of $G$ beyond torsion-freeness is needed.
If the condition holds, every multi-term polynomial is a sum of exactly two irreducibles except for the stated exceptions: binomials with one unit coefficient and trinomials whose three coefficients are units.
For group series semidomains over such coefficient semidomains, the corresponding weak Goldbach law holds with at most three irreducibles whenever $G$ has well-ordered bounded-below subsets, and in particular whenever $G$ is a finitely generated torsion-free abelian group.
The condition is necessary: if $S$ contains an additive atom that is not a unit, then some trinomial in $S[G]$ fails to be a sum of two irreducibles, so the Goldbach property fails.
The example $S_{pq}$ shows the hypotheses cover semidomains that are additively Furstenberg but not additively atomic, so the theorem is not limited to atomic coefficient semirings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The converse part of the proof builds its obstruction from a single non-unit additive atom repeated across several exponents, which suggests a quantitative strengthening one could test: in the failure case, the minimum number of irreducibles needed for a multi-term polynomial is controlled by the number of terms, not by the group.
Because the monolithicity lemmas rely only on a compatible linear order and the absence of additive inverses, a natural extension would be to ordered cancellative semigroups rather than groups, or to non-abelian ordered groups, where the notion of support still makes sense.
The constant $(1+\sqrt5)/2$ in Example 3.7 looks like a genuine boundary: testing values $q/p$ at or below the golden ratio should show where the additively Furstenberg property fails, marking the exact scope of the theorem.
If the equivalence turns out to be robust, it gives a cheap certificate for Goldbach behavior in semiring constructions: verify one equality in the coefficient object, and the entire semidomain inherits the property.

Original abstract

A semidomain is a subsemiring of an integral domain. We call a semidomain $S$ additively reduced if $0$ is the only invertible element of the monoid $(S, +)$, while we say that $S$ is additively Furstenberg if every non-invertible element of $(S,+)$ can be expressed as the sum of an atom and an element of $S$. In this paper, we study a variant of the Goldbach conjecture within the framework of group semidomains $S[G]$ and group series semidomains $S[\![G]\!]$, where $S$ is both an additively reduced and additively Furstenberg semidomain and $G$ is a torsion-free abelian group. In particular, we show that every non-constant polynomial expression in $S[G]$ can be written as the sum of at most two irreducibles if and only if the condition $\mathscr{A}_+(S) = S^\times$ holds.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,902 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.36 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.94 · compares with prior results 0.07 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

A semidomain is a subsemiring of an integral domain. We call a semidomain $S$ additively reduced if $0$ is the only invertible element of the monoid $(S, +)$, while we say that $S$ is additively Furstenberg if every non-invertible element of $(S,+)$ can be expressed as the sum of an atom and an element of $S$. In this paper, we study a variant of the Goldbach conjecture within the framework of group semidomains $S[G]$ and group series semidomains $S[\![G]\!]$, where $S$ is both an additively reduced and additively Furstenberg semidomain and $G$ is a torsion-free abelian group. In particular, we show that every non-constant polynomial expression in $S[G]$ can be written as the sum of at most two irreducibles if and only if the condition $\mathscr{A}_+(S) = S^\times$ holds.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A semidomain is a subsemiring of an integral domain.	background 1.00	0.04	0.01	asserted 1.00
s1	We call a semidomain $S$ additively reduced if $0$ is the only invertible element of the monoid $(S, +)$, while we say that $S$ is additively Furstenberg if every non-invertible element of $(S,+)$ can be expressed as the sum of an atom and an element of $S$.	background 1.00	0.14	0.02	asserted 0.92
s2	In this paper, we study a variant of the Goldbach conjecture within the framework of group semidomains $S[G]$ and group series semidomains $S[\![G]\!]$, where $S$ is both an additively reduced and additively Furstenberg semidomain and $G$ is a torsion-free abelian group.	contribution 0.98	0.55	0.10	asserted 0.77
s3	In particular, we show that every non-constant polynomial expression in $S[G]$ can be written as the sum of at most two irreducibles if and only if the condition $\mathscr{A}_+(S) = S^\times$ holds.	result 0.99	0.95	0.79	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

arXiv listing abstract v2

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levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (16)

[10]

N. Kaplan and H. Polo: A Goldbach theorem for Laurent series semidomains , arXiv:https://arxiv.org/abs/2312.14888

View on Pith arXiv
[13]

S. Liao and H. Polo: A Goldbach theorem for Laurent polynomials with positive in teger coeﬃcients , Amer. Math. Monthly 131 (2024) 704–711

View on Pith 2024
[1]

A. O. Bender: Representing an element in Fq[t] as the sum of two irreducibles , Mathematika 60 (2014) 166–182

View on Pith 2014
[2]

F. Campanini and A. Facchini: Factorizations of polynomials with integral non-negative coeﬃ- cients, Semigroup Forum 99 (2019) 317–332

View on Pith 2019
[3]

M. Car and L. H. Gallardo: Representation of a polynomial as the sum of an irreducible p olynomial and a square-free polynomial , Acta Arith. 197 (2021) 293–309

View on Pith 2021
[4]

S. T. Chapman and H. Polo: Arithmetic of additively reduced monoid semidomains , Semigroup Forum 107 (2023) 40–59

View on Pith 2023
[5]

G. W. Eﬃnger and D. R. Hayes: A complete solution to the polynomial 3-primes problem , Bull. Amer. Math. Soc. 24 (1991) 363–369

View on Pith 1991
[6]

H. Fox, A. Goel, and S. Liao: Arithmetic of semisubtractive semidomains , J. Algebra Appl. (Online Ready) https://doi.org/10.1142/S0219498825501634

View on Pith doi:10.1142/s0219498825501634
[7]

J. S. Golan: Semirings and their Applications , Kluwer Academic Publishers, 1999

View on Pith 1999
[8]

F. Gotti and H. Polo: On the arithmetic of polynomial semidomains , Forum Math. (2023) 35 1179–1197

View on Pith 2023
[9]

D. R. Hayes: A Goldbach theorem for polynomials with integral coeﬃcient s, Amer. Math. Monthly 72 (1965) 45–46

View on Pith 1965
[11]

M. Kozek: An asymptotic formula for Goldbach’s conjecture with monic polynomials in Z[x], Amer. Math. Monthly 117 (2010) 365–369

View on Pith 2010
[12]

F. Levi: Arithmetische gesetze im gebiete diskreter gruppen , Rend. Circ. Matem. Palermo 35 (1913) 225–236

View on Pith 1913
[14]

E. Paran: Twin-prime and Goldbach theorems for Z/llbracketx/rrbracket, J. Number Theory 213 (2020) 453–461

View on Pith 2020
[15]

P. Pollack: On polynomial rings with a Goldbach property , Amer. Math. Monthly 118 (2011) 71–77

View on Pith 2011
[16]

F. Saidak: On Goldbach’s conjecture for integer polynomials , Amer. Math. Monthly 113 (2006) 541–545

View on Pith 2006

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites

Builds on 1 paper

To do so, we extend several key results from [ref] .

2312.14888 · A Weak Goldbach Theorem for Group Series Semidomains · 1.00

Uses the method or tool of 1 paper

Our approach closely follows key ideas presented in [ref] .

2312.14888 · A Goldbach Theorem for Group Semidomains · 0.97

Cites as background 1 paper

In [ref] , the authors studied analogues of the Goldbach conjecture for polynomials with coefficients in an additively reduced and additively atomic semidomain.

2312.14888 · A Goldbach Theorem for Group Semidomains · 0.96

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [10]

N. Kaplan and H. Polo: A Goldbach theorem for Laurent series semidomains , arXiv:https://arxiv.org/abs/2312.14888

View on Pith arXiv

[2] [13]

S. Liao and H. Polo: A Goldbach theorem for Laurent polynomials with positive in teger coeﬃcients , Amer. Math. Monthly 131 (2024) 704–711

View on Pith 2024

[3] [1]

A. O. Bender: Representing an element in Fq[t] as the sum of two irreducibles , Mathematika 60 (2014) 166–182

View on Pith 2014

[4] [2]

F. Campanini and A. Facchini: Factorizations of polynomials with integral non-negative coeﬃ- cients, Semigroup Forum 99 (2019) 317–332

View on Pith 2019

[5] [3]

M. Car and L. H. Gallardo: Representation of a polynomial as the sum of an irreducible p olynomial and a square-free polynomial , Acta Arith. 197 (2021) 293–309

View on Pith 2021

[6] [4]

S. T. Chapman and H. Polo: Arithmetic of additively reduced monoid semidomains , Semigroup Forum 107 (2023) 40–59

View on Pith 2023

[7] [5]

G. W. Eﬃnger and D. R. Hayes: A complete solution to the polynomial 3-primes problem , Bull. Amer. Math. Soc. 24 (1991) 363–369

View on Pith 1991

[8] [6]

H. Fox, A. Goel, and S. Liao: Arithmetic of semisubtractive semidomains , J. Algebra Appl. (Online Ready) https://doi.org/10.1142/S0219498825501634

View on Pith doi:10.1142/s0219498825501634

[9] [7]

J. S. Golan: Semirings and their Applications , Kluwer Academic Publishers, 1999

View on Pith 1999

[10] [8]

F. Gotti and H. Polo: On the arithmetic of polynomial semidomains , Forum Math. (2023) 35 1179–1197

View on Pith 2023

[11] [9]

D. R. Hayes: A Goldbach theorem for polynomials with integral coeﬃcient s, Amer. Math. Monthly 72 (1965) 45–46

View on Pith 1965

[12] [11]

M. Kozek: An asymptotic formula for Goldbach’s conjecture with monic polynomials in Z[x], Amer. Math. Monthly 117 (2010) 365–369

View on Pith 2010

[13] [12]

F. Levi: Arithmetische gesetze im gebiete diskreter gruppen , Rend. Circ. Matem. Palermo 35 (1913) 225–236

View on Pith 1913

[14] [14]

E. Paran: Twin-prime and Goldbach theorems for Z/llbracketx/rrbracket, J. Number Theory 213 (2020) 453–461

View on Pith 2020

[15] [15]

P. Pollack: On polynomial rings with a Goldbach property , Amer. Math. Monthly 118 (2011) 71–77

View on Pith 2011

[16] [16]

F. Saidak: On Goldbach’s conjecture for integer polynomials , Amer. Math. Monthly 113 (2006) 541–545

View on Pith 2006