Monomial algebraic and algebroid curves and their Lipschitz saturation over the ground ring

Anne Fr\"uhbis-Kr\"uger; Thiago da Silva

REVIEW 3 major objections 4 minor 23 references

Monomial algebraic and algebroid curves and their Lipschitz saturation over the ground ring

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The Lipschitz saturation of a monomial curve algebra is explicitly determined by the gap structure of its numerical semigroup.

desk verdict The main theorem is false: Proposition 2.5 confuses integral closure with ideal membership, and Γ=<5,7> gives a concrete counterexample. read the letter →

arxiv 2411.17579 v1 pith:GM7T3XEL submitted 2024-11-26 math.AC

Anne Frühbis-Krüger , Thiago da Silva This is my paper · ORCID

classification math.AC MSC 13B2214H20

keywords Lipschitzsaturationrelativenumericalsemigroupmonomialcurvealgebroidpowerseriesringbi-Lipschitzequisingularityintegralclosure

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to turn the Lipschitz saturation of a monomial curve algebra into a computation rather than an abstract closure operation. For a subalgebra $A = R[t^{\gamma_1},\ldots,t^{\gamma_n}]$ inside a larger ring $B$ of power series in $t$, it claims that the saturation $A^*_{B,R}$ is obtained by adjoining a finite set of monomials $t^\ell$, with the exponents determined solely by the numerical semigroup $\Gamma$ generated by the $\gamma_i$. The description is uniform across three natural choices of $B$: polynomials, all power series, and convergent analytic series over $\mathbb{R}$ or $\mathbb{C}$. Because the extra monomials are explicit, the result reduces a question about tensor products and integral closures to a semigroup calculation, which matters since Lipschitz saturation is the algebraic counterpart of bi-Lipschitz equisingularity of curve germs.

What carries the argument

The load-bearing object is the relative Lipschitz saturation $A^*_{B,R}$, defined for $R$-algebras $A \subseteq B$ through the diagonal map $\Delta(x) = x \otimes 1 - 1 \otimes x$ and the kernel of the canonical map $B \otimes_R B \to B \otimes_A B$: an element of $B$ is saturated when $\Delta(x)$ lies in that kernel. The proof's engine is a pair of semigroup facts: a divisibility and induction lemma shows that once monomials of degrees $\alpha$ and $\alpha+d$ are saturated, every monomial of degree $\alpha + sd$ is saturated; and a root-of-unity argument shows that in a saturated series no coefficient can sit in a gap degree not divisible by the relevant partial gcd, provided the residue characteristic does not divide $\gamma_1$. These forces combine with the gap structure of $\Gamma$ to leave exactly the listed monomials.

What would settle it

Take $R = \mathbb{F}_2$, $\Gamma = \langle 2, 3 \rangle$, $B = \mathbb{F}_2[t]$, and $A = \mathbb{F}_2[t^2,t^3]$, then compute $A^*_{B,R}$ directly from the definition; here $\operatorname{char} R$ divides $\gamma_1$, so the theorem is not asserted, and comparing the result with the predicted $A[t^{L(r)}]$ would show whether the characteristic hypothesis is essential.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 3.14: under a noetherian $\gamma_1$-nice ground ring $R$ and the assumption that $A$ is $(\Gamma,B)$-closed, the Lipschitz saturation equals $A[t^{L(r)}]$, where $r$ is the first index with $d_r = \gcd(\gamma_1,\ldots,\gamma_r)=1$, and $L(r)$ is the union of the intermediate gap sets $L_j = \{\ell \in G(\Gamma) : \gamma_j < \ell < \gamma_{j+1},\ d_j \mid \ell,\ \ell \leq \gamma_j + \gamma_1 - 1\}$ together with the terminal interval $\tilde L(r) = \{\ell \in G(\Gamma) : \gamma_r + 1 \leq \ell \leq \gamma_r + \gamma_1 - 1\}$. The same equality has explicit $R$-module and $A$-module versions. Thus the saturation is a finitely generated $A$-algebra with an explicit monomial presentation, and in the two-generator case with $\gamma_1 = 2$ the saturation collapses to $A$ itself.

Load-bearing premise

The proof relies on the ground ring being noetherian and on a property called $(\Gamma,B)$-closedness—informally, that removing all gap-degree terms from an element of $B$ leaves an element of $A$—which is verified only for specific choices of $B$ and is not characterized in general.

Editorial extensions

If this is right

Under the theorem's hypotheses, the Lipschitz saturation of a monomial curve algebra is a finitely generated $A$-algebra with an explicitly listed monomial generating set.
For the polynomial, formal power series, and convergent analytic choices of $B$, the saturation can be computed algorithmically from the semigroup $\Gamma$ and the partial gcds $d_j$.
When $\Gamma = \langle 2, \gamma_2 \rangle$ with $\gamma_2$ odd and the characteristic condition holds, every $(\Gamma,B)$-closed $A$ is already Lipschitz saturated, with no extra monomials needed.
The worked examples show that the final minimal generating set is often smaller than the raw gap list, since products of added generators may cover some listed degrees.

Reading between the lines

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

A likely next step is to characterize $(\Gamma,B)$-closedness in terms of the conductor of $\Gamma$ or the coefficient module of $B$, which would extend the theorem beyond the three choices of $B$ checked in the paper.
Because Lipschitz saturation sits between $A$ and its normalization, the same gap-set formula may yield a direct algorithm for the integral closure or normalization of monomial algebroid curves.
The characteristic assumption enters only through the root-of-unity step, so small examples with $\operatorname{char} R$ dividing $\gamma_1$ would show whether that hypothesis is essential or merely an artifact of the proof.
The finite set $L(r)$ could be packaged as a new semigroup invariant, analogous to the Apéry set, and compared with other invariants of numerical semigroups.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the bottom line: the central claim does not survive contact with a simple example. Proposition 2.5 tries to show z1^{αm+d}-z2^{αm+d} lies in the ideal generated by the z1^{αi}-z2^{αi}, but the proof only establishes membership in the integral closure, using the valuative criterion. Those are different. Take T=k[x,y], α=(5,7), d=1. Then x^8-y^8 has no homogeneous degree-8 representation in (x^5-y^5,x^7-y^7); the coefficient equations force a0+b0=1 and a0=b0=0. So Corollary 2.6 and Proposition 3.3 are false, and Theorem 3.14 fails for Γ=<5,7> with B=k[t], A=k[t^5,t^7]: the predicted t^8 is not in A^*_{B,R}.

That is the load-bearing flaw. I want to give credit where it is due: the overall strategy—describing the saturation via the semigroup and partial gcds—is natural, and the relative setting over a general ring is a reasonable extension. The early lemmas (2.1–2.4) are correct, and the appendix examples show the formula in action. The (Γ,B)-closed notion is a sensible condition, though not characterized.

The secondary issues are real but minor by comparison. Observation 3.1 is stated without proof and relies on passing to the completed tensor product; that needs a careful argument. The appendix examples would also be more convincing with a reproducible script, but they are not the problem.

If the paper were correct, it would be a useful tool for the Lipschitz-saturation community. As it stands, the main theorem is false, and the error is not a gap you can patch with a remark—it is a counterexample. The authors may be able to recover something by rephrasing the result in terms of integral closure, but that would be a different paper.

For peer review: send it out if you want, because the mistake is instructive and the topic is legitimate. But the referee report should be a rejection or a demand for a major reworking. I would not cite it in its current form.

Referee Report

3 major / 4 minor

Summary. The manuscript studies the relative Lipschitz saturation A*_{B,R} of a monomial subalgebra A ⊆ R[[t^{γ_1},...,t^{γ_n}]] inside an R-subalgebra B ⊆ R[[t]] containing R[t]. Its main result, Theorem 3.14, claims an explicit description A*_{B,R} = A[t^{L(r)}] in terms of the numerical semigroup Γ = ⟨γ_1,...,γ_n⟩, under the assumptions that R is a noetherian γ_1-nice ring and that A is (Γ,B)-closed. Section 2 develops algebraic membership results for binomial differences, and Section 3 uses those results, together with Proposition 3.4, to prove the theorem. The appendix lists computed examples for several semigroups.

Significance. If correct, the result would give a useful and explicit computation of relative Lipschitz saturation for monomial curves over general ground rings, extending earlier analytic results. The paper introduces convenient notation involving partial gcds and gap sets, and the reverse-inclusion part of the argument is plausible. However, the central claim is false: the proof of Proposition 2.5 mistakes a valuative criterion for integral closure for a criterion of ideal membership, and a concrete counterexample satisfying all hypotheses of Theorem 3.14 breaks the main result. The paper therefore does not establish its advertised description.

major comments (3)

[§2, Proposition 2.5] Proposition 2.5 is false. The proof invokes the Valuative Criterion [22, Theorem 6.8.3], which characterizes membership in the integral closure of an ideal, not membership in the ideal itself; the DVR argument establishes at most h ∈ \overline{I}. In the noetherian ring T = k[x,y] with z1 = x, z2 = y, α1 = 5, α2 = 7, d = 1, the proposition claims x^8 − y^8 ∈ I = (x^5 − y^5, x^7 − y^7). Since I is homogeneous, any degree-8 representation must take the form A(x^5 − y^5) + B(x^7 − y^7) with deg A = 3 and deg B = 1. Writing A = a0 x^3 + a1 x^2 y + a2 x y^2 + a3 y^3 and B = b0 x + b1 y, the coefficients of x^8, x^6 y^2, x^3 y^5, and y^8 give a0 + b0 = 1, a2 = 0, −a0 = 0, and −a3 − b1 = −1, so a0 = 0 and b0 = 0, contradicting a0 + b0 = 1. Hence x^8 − y^8 ∉ I, and Proposition 2.5 is false.
[§2 Corollary 2.6 and §3 Theorem 3.14] The failure of Proposition 2.5 invalidates Corollary 2.6 and Proposition 3.3, and consequently Theorem 3.14. For a concrete disproof of the main theorem, take a field k with char k ≠ 5, R = k, B = k[t], A = k[t^5, t^7], and Γ = ⟨5,7⟩. All hypotheses of Theorem 3.14 hold: Γ is coprime, 5 ∤ 7, d2 = 1, R is noetherian and γ1-nice, and A is (Γ,B)-closed by Proposition 3.8(a). Corollary 2.4, which is correct, gives ker φ = ⟨t1^5 − t2^5, t1^7 − t2^7⟩ in k[t1,t2]. If t^8 belonged to A*_{B,R}, then t1^8 − t2^8 would lie in that ideal, but the degree argument in the previous comment shows it does not. Thus t^8 ∉ A*_{B,R}, whereas Theorem 3.14(c) predicts t^8 ∈ A[t^{L(2)}] because 8 ∈ L~(2) ⊆ L(2). The main theorem is therefore false.
[§3, Observation 3.1] Observation 3.1 is asserted without proof. It claims that for C ⊆ k[[t]] and A′ ⊆ k[[x1,...,xs]] ∩ C, the kernel of C ⊗_k C → C ⊗_{A′} C is contained in the ideal ⟨Δ(x1),...,Δ(xs)⟩ in k[[t1,t2]]. This requires passing to the completed tensor product, justifying injectivity of C ⊗_k C → k[[t1,t2]], and proving that the image of the kernel is contained in the closed ideal generated by the Δ(xi). None of these steps is given, yet Proposition 3.4 and consequently Lemma 3.12 rely on this inclusion.

minor comments (4)

[Global] The manuscript contains a 'DRAFT - GSN - 2022' header and numerous OCR-type artifacts (e.g., '/shortrightarrow' symbols and garbled ligatures); these should be cleaned before any submission.
[§3, before Lemma 3.12] The notation L_j is redefined as L_j := {ℓ ∈ L_j | ℓ ≤ γ_j + γ1 − 1} after L_j was already defined in Section 3; this reuse is confusing and a different symbol, such as L_j^#, would be clearer.
[Appendix A] The appendix examples derive their claimed equalities from Theorem 3.14; since the theorem is false, these examples should be rechecked independently of the theorem before being presented as illustrations.
[Theorem 3.14(c)] Part (c) writes A*_{B,R} = A[t^{L(r)}] = A[t^{L(r)}], but the two displayed sets are not defined identically (the truncated L(r) versus the full L(r)); the intended relationship should be stated precisely with distinct symbols.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Lipschitz saturation is defined externally via Lipman and the semigroup formula is derived, not fitted or imported by self-citation.

full rationale

The paper's central claim, Theorem 3.14, is a genuine derivation from Lipman's definition of relative Lipschitz saturation together with standard commutative algebra tools. The definition of A*_{B,R} is taken from Lipman [17] as an external input and is not defined in terms of the semigroup formula L(r); therefore the equality A*_{B,R} = A[t^{L(r)}] cannot reduce to its own definition. The semigroup description is obtained by proving two inclusions: Corollary 2.6 and Proposition 3.3 supply the containment A[t^{L(r)}] subset A*_{B,R}, while Lemma 3.11 and Lemma 3.12 supply the reverse containment. These arguments use the valuative criterion from Swanson-Huneke and elementary manipulations of differences of monomials, not the target formula. The paper does cite prior work by one of the authors, but those citations appear in the introduction as context and are not load-bearing for the proof of Theorem 3.14. The unproved containment in Observation 3.1 and the correctness of Proposition 2.5 are potential mathematical gaps, but a false or unproved step is not circularity: the step is not equivalent to the theorem by construction, and no parameter is fitted and then renamed as a prediction. Hence the derivation is self-contained with respect to circularity concerns.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces technical conditions gamma_1-nice and (Gamma,B)-closed as hypotheses, but these are not invented entities: they are conditions on the ring R and the pair (Gamma,B). The central claim rests on standard commutative algebra: noetherianity, integral closure, the valuative criterion, and the structure of completed tensor products. The (Gamma,B)-closed assumption is the most restrictive, since it is not characterized for general B. The proof relies on the standard identification of the kernel of the canonical map in the polynomial case, a fact the paper extends to power series rings with only a sketch.

assumptions (6)

domain assumption R is a commutative unital ring, and in the main theorem R is noetherian and gamma_1-nice.
The paper assumes all rings are commutative and unitary, and Theorem 3.14 explicitly assumes R is noetherian and gamma_1-nice, which restricts the applicability to rings of characteristic not dividing gamma_1.
domain assumption The ambient ring B contains R[t] and A is an R-subalgebra of B ∩ R[[t^{gamma_1},...,t^{gamma_n}]] containing R[t^{gamma_1},...,t^{gamma_n}].
The setup of the main theorem is stated in the introduction and used throughout; the Lipschitz saturation is only defined in this context.
domain assumption A is (Gamma,B)-closed.
Definition 3.7 introduces the (Gamma,B)-closed property, and Lemma 3.11 and Theorem 3.14 assume it. Proposition 3.8 gives three sufficient cases, but the property is not characterized in general.
standard math The valuative criterion for integral closure (Swanson-Huneke, Theorem 6.8.3) is invoked in Proposition 2.5 and Proposition 3.4.
The paper cites [22, Theorem 6.8.3] and relies on the valuative criterion to check containment of an element in an ideal via discrete valuation rings. This is a standard theorem.
domain assumption For a k-subalgebra C of k[[t]], the tensor product C ⊗_k C embeds into k[[t]] ⊗_k k[[t]] and then into the completed tensor product k[[t]] ≅ k[[t1,t2]].
Observation 3.1 relies on the flatness of k-algebras and the identification of the completed tensor product with k[[t1,t2]]. This step is load-bearing for Proposition 3.4 and is stated without proof.
domain assumption The ideal containment in the completed tensor product ring implies the needed coefficient vanishing in the original ring.
Proposition 3.4 uses containment in k[[t1,t2]] to conclude that certain coefficients vanish. The transfer from the completed ring back to the algebraic tensor product is not fully justified.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Monomial algebraic and algebroid curves and their Lipschitz saturation over the ground ring." pith.science (2026). https://pith.science/paper/GM7T3XEL

@misc{pith2026241117579,
  author       = {Pith},
  title        = {Pith review of: Monomial algebraic and algebroid curves and their Lipschitz saturation over the ground ring},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GM7T3XEL}},
  note         = {Machine review of arXiv:2411.17579}
}

abstract

In this article, we provide an explicit description of the Lipschitz saturation $A^*_{B,R}$ of a subalgebra $A\subseteq R[[t^{\gamma_1},\ldots, t^{\gamma_n}]]$ over a ring $R$ in terms of the numerical semigroup $\Gamma = \langle\gamma_{1},\ldots,\gamma_{n}\rangle$ with mild conditions imposed on $R$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Altman and S

A. Altman and S. Kleiman, A term of commutative algebra , Cambridge, USA, 2021

work page 2021
[2]

Relative Lipschitz saturation of complex algebraic varieties

F. Bernard, Relative Lipschitz saturation of complex algebraic variet ies, arXiv:2312.14517, (2023)

work page Pith review arXiv 2023
[3]

Bourbaki, Elements of Mathematics: Algebra II (Chapters 4-7) , (Translated by P

N. Bourbaki, Elements of Mathematics: Algebra II (Chapters 4-7) , (Translated by P. M. Cohn and J. Howie), Springer-Verlag, (1981)

work page 1981
[4]

da Silva, Categorical aspects of Gaﬀney’s double structure of a modul e, Matem´ atica Contemporˆ anea, Vol

T. da Silva, Categorical aspects of Gaﬀney’s double structure of a modul e, Matem´ atica Contemporˆ anea, Vol. 53, 213-232, (2023)

work page 2023
[5]

da Silva and M

T. da Silva and M. Ribeiro, Universally injective and integral contractions on relati ve Lipschitz saturation of algebras, Journal of Algebra, Vol. 662, 902-922, (2025)

work page 2025
[6]

da Silva and N

T. da Silva and N. G. Grulha Jr. and M. Pereira, The Bi-Lipschitz Equisingularity of Essentially Iso- lated Determinantal Singularities , Bulletin of the Brazilian Mathematical Society, New Series, 49, 637- 645, (2018). 14 DRAFT - GSN - 2022

work page 2018
[7]

da Silva and N

T. da Silva and N. G. Grulha Jr. and M. Pereira, Real and Complex Integral Closure, Lipschitz equisingu- larity and applications on square matrices , Journal of Singularities, Vol. 22, 215-226, (2020)

work page 2020
[8]

Delgado, P

M. Delgado, P. A. Garc ´ ıa-S´ anchez and J.J. Morais,numericalsgps – A package for numerical semigroups , Version 1.3.1, Centro de Matem´ atica da Universidade do Porto, Por tugal and Universidad de Granada, Spain https://docs.gap-system.org/pkg/numericalsgps/doc/c hap0.html. (2005–2015)

work page 2005

Show all 23 references

[9]

The GAP ∼Group, GAP – Groups, Algorithms, and Programming, Version 4.13.1 ; https://www.gap-system.org, (2024)

2024
[10]

Fernandes and M

A. Fernandes and M. A. S. Ruas, Bilipschitz determinacy of quasihomogeneous germs , Glasgow Mathemat- ical Journal, 46 (1), 77-82 (2004)

2004
[11]

Fernandes and M

A. Fernandes and M. A. S. Ruas, Rigidity of bi-Lipschitz equivalence of weighted homogene ous function- germs in the plane , Proc. Amer. Math. Soc. 141, 1125-1133 (2013)

2013
[12]

Gaﬀney, Bi-Lipschitz equivalence, integral closure and invariants, Real and complex singularities, 125137, London Math

T. Gaﬀney, Bi-Lipschitz equivalence, integral closure and invariants, Real and complex singularities, 125137, London Math. Soc. Lecture Note Ser., 380, Cambridge Univ. Press , Cambridge, (2010)

2010
[13]

Gaﬀney, The genericity of the inﬁnitesimal Lipschitz condition for hypersurfaces, Journal of Singularities, Vol

T. Gaﬀney, The genericity of the inﬁnitesimal Lipschitz condition for hypersurfaces, Journal of Singularities, Vol. 10, 108-123, (2015)

2015
[14]

Gaﬀney and T

T. Gaﬀney and T. da Silva, Inﬁnitesimal Lipschitz conditions on a family of analytic v arieties: genericity and necessity, S˜ ao Paulo J. Math. Sci., https://doi.org/10.1007/s40863-024-0 0452-5, (2024)

2024 doi
[15]

Gaﬀney and T

T. Gaﬀney and T. da Silva, The generic equivalence among the Lipschitz saturations of a sheaf of modules , Research in the Mathematical Sciences, Vol. 11, (32), (2024)

2024
[16]

Lejeune-Jalabert and B

M. Lejeune-Jalabert and B. Teissier, Clˆ oture int´ egrale des id´ eaux et equisingularit´ e, S´ eminaire Lejeune- Teissier, Centre de Math´ ematiques ´Ecole Polytechnique, (1974) Publ. Inst. Fourier St. Martin d’Heres , F-38402, (1975)

1974
[17]

Lipman, Relative Lipschitz-saturation, Amer

J. Lipman, Relative Lipschitz-saturation, Amer. J. Math. 97, no. 3, 791-813, (1975)

1975
[18]

OSCAR – Open Source Computer Algebra Research system, Ver sion 1.0.0, The OSCAR Team, (https://www.oscar-system.org), (2024)

2024
[19]

Pham and B

F. Pham and B. Teissier, Fractions lipschitziennes d’une alg´ ebre analytique comp lexe et saturation de Zariski , Centre de Math´ ematiques de l’Ecole Polytechnique (Paris), http://people.math.jussieu.fr/teissier/old-papers.html, (1969)

1969
[20]

Pham, Fractions lipschitziennes et saturation de Zariski des alg ` ebres analytiques complexes

F. Pham, Fractions lipschitziennes et saturation de Zariski des alg ` ebres analytiques complexes. Expos´ e d’un travail fait avec Bernard Teissier. Fractions lipschitziennes d’une a lg` ebre analytique complexe et saturation de Zariski, Centre Math. l ´Ecole Polytech., Paris,...

1969
[21]

Rosales and P.A

J.C. Rosales and P.A. Garcia-S´ anchez, Numerical Semigroups, (First. ed.). New York: Springer, (2009)

2009
[22]

Swanson and C

I. Swanson and C. Huneke, Integral closure of ideals, rings and modules, London Mathematical Society, Lecture Notes Series 336, Cambridge, UK, (2006)

2006
[23]

Zariski, General Theory of Saturation and of Saturated Local Rings

O. Zariski, General Theory of Saturation and of Saturated Local Rings. I I. Saturated local rings of dimension 1 Amer. J. Math. 93, 872-964, (1971). Anne Fr ¨uhbis-Kr¨uger (Carl von Ossietzky Universit ¨at Oldenburg) anne.fruehbis-krueger@uni-oldenburg.de Thiago da Silva (Fede...

1971

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Altman and S

A. Altman and S. Kleiman, A term of commutative algebra , Cambridge, USA, 2021

work page 2021

[2] [2]

Relative Lipschitz saturation of complex algebraic varieties

F. Bernard, Relative Lipschitz saturation of complex algebraic variet ies, arXiv:2312.14517, (2023)

work page Pith review arXiv 2023

[3] [3]

Bourbaki, Elements of Mathematics: Algebra II (Chapters 4-7) , (Translated by P

N. Bourbaki, Elements of Mathematics: Algebra II (Chapters 4-7) , (Translated by P. M. Cohn and J. Howie), Springer-Verlag, (1981)

work page 1981

[4] [4]

da Silva, Categorical aspects of Gaﬀney’s double structure of a modul e, Matem´ atica Contemporˆ anea, Vol

T. da Silva, Categorical aspects of Gaﬀney’s double structure of a modul e, Matem´ atica Contemporˆ anea, Vol. 53, 213-232, (2023)

work page 2023

[5] [5]

da Silva and M

T. da Silva and M. Ribeiro, Universally injective and integral contractions on relati ve Lipschitz saturation of algebras, Journal of Algebra, Vol. 662, 902-922, (2025)

work page 2025

[6] [6]

da Silva and N

T. da Silva and N. G. Grulha Jr. and M. Pereira, The Bi-Lipschitz Equisingularity of Essentially Iso- lated Determinantal Singularities , Bulletin of the Brazilian Mathematical Society, New Series, 49, 637- 645, (2018). 14 DRAFT - GSN - 2022

work page 2018

[7] [7]

da Silva and N

T. da Silva and N. G. Grulha Jr. and M. Pereira, Real and Complex Integral Closure, Lipschitz equisingu- larity and applications on square matrices , Journal of Singularities, Vol. 22, 215-226, (2020)

work page 2020

[8] [8]

Delgado, P

M. Delgado, P. A. Garc ´ ıa-S´ anchez and J.J. Morais,numericalsgps – A package for numerical semigroups , Version 1.3.1, Centro de Matem´ atica da Universidade do Porto, Por tugal and Universidad de Granada, Spain https://docs.gap-system.org/pkg/numericalsgps/doc/c hap0.html. (2005–2015)

work page 2005