{"id":"0e19f75f-dbf6-413e-babe-7cf26ac8bfa7","arxiv_id":"2411.10834","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Gauss-Borel factorization of a rectangular moment matrix yields the first systematic framework for mixed multiple orthogonal Laurent polynomials on the unit circle, with Christoffel and Geronimus perturbation formulas.","lead":"This paper builds a theory of mixed multiple orthogonal Laurent polynomials on the unit circle using Gauss-Borel factorization of a matrix of moments, and derives recurrences, kernels, and perturbation formulas. A generalist should care because it connects two active areas, multiple orthogonal polynomials and orthogonal polynomials on the unit circle, and provides tools for spectral theory and integrable systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Geronimus singular part placed off T vanishes from \\tilde M; Proposition 4.7's mass-dependent formulas are not connected to \\tilde B,\\tilde A as defined.","rationale":"The reader's verdict was CONDITIONAL, with the weakest assumption identified as the existence of the Gauss–Borel factorization plus nonzero determinant conditions, and a note that the Geronimus singular part is off-circle. My stress-test isolates a stronger, more specific problem in the Geronimus section: the singular part, as defined, is invisible to the moment matrix \\tilde M because it is supported off T. This makes \\tilde B and \\tilde A independent of the singular part, while Proposition 4.7's formulas depend on the singular part's masses. That is an internal inconsistency, not merely an unproven genericity condition. The Christoffel section (Propositions 3.2 and 3.3) may be salvageable—the columnwise connection formula is correct even if the first displayed equality in Proposition 3.2 appears to be a notational slip—and the core recurrence construction is solid conditional on factorization. But the Geronimus results, which are advertised as a central contribution, are not well-founded as written. A concrete small-scale computation with p=q=1 and off-circle zeros can decisively confirm or refute this concern. If the test shows the mass terms do not cancel, the Geronimus formulas must be rejected or substantially revised, which is why I recommend moving from CONDITIONAL to REJECT. I agree with the reader's flagging of the off-circle departure, but I differ in that I see it as a fundamental contradiction rather than a secondary caveat.","tokens_in":54028,"tokens_out":27998,"duration_ms":249819,"concrete_test":"Take p=q=1, dμ = dθ/(2π) on T, and W(z)=c(z-2)(z-1/2)/z (two simple zeros off T). For m∈{0,1}, define d\\tilde μ_m = W^{-1} dμ + m δ(z-2). Compute the truncated moment matrix \\tilde M_3(m)=∮_T Z_3(z) d\\tilde μ_m(z) Z_3^T(z^{-1}) explicitly; it should be identical for m=0 and m=1. Compute the corresponding Gauss–Borel factors and \\tilde A_2^{(1)}(z^{-1}) for both m. Then evaluate the Christoffel–Geronimus formula in Proposition 4.7 with the same data and masses m. If the formula gives different results for m=0 and m=1 while the actual \\tilde A is unchanged, Proposition 4.7 is invalid. If the results coincide, the mass dependence miraculously cancels and the paper would be vindicated.","verdict_should_be":"REJECT","load_bearing_attack":"The Geronimus section (Section 4.2) defines the perturbed measure in (94) as d\\tilde μ = W^{-1} dμ + dμ_s, where dμ_s is a sum of Dirac deltas supported at the 2dq zeros of W[ q]. The paper explicitly states that these zeros are not located on T (Remark after (96)). Consequently, for every integral over T, ∫_T f(z) dμ_s(z) = 0. Therefore the moment matrix \\tilde M in Proposition 4.1, defined as ∮_T Z d\\tilde μ Z^T(z^{-1}), is identical to the moment matrix of W^{-1} dμ alone and is completely independent of the masses m_{b,a,j}. It follows that the Gauss–Borel factors \\tilde L, \\tilde U, and hence the Geronimus polynomials \\tilde B, \\tilde A from (84), are independent of the singular part. Yet Proposition 4.7 derives an explicit Christoffel–Geronimus formula for \\tilde A whose entries F^{(b)}_{n,j} depend directly on the masses m_{b,a,j}. Unless those mass terms cancel identically—which is neither shown nor plausible—the formula cannot describe the \\tilde A obtained from the stated moment matrix. This is not a missing proof or a mild assumption; it is an internal inconsistency between the definition of the measure and the integral used to define the moment matrix. The issue could be repaired by placing the singular part on T (so it contributes to the moments), but the paper chooses off-circle zeros and does not address this contradiction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript constructs a theory of mixed multiple orthogonal Laurent polynomials on the unit circle (MOLPUC) from the Gauss–Borel factorization of a semi-infinite CMV moment matrix associated with a q×p matrix of measures. It introduces left and right families of Laurent polynomial matrices B(z) and A(z^{-1}), proves matrix biorthogonality and entrywise mixed orthogonality relations, states degree bounds, and derives banded recurrence relations T B(z)=zB(z) and \\bar A(z^{-1})T=z\\bar A(z^{-1}), together with Szegő-type recurrences and Christoffel–Darboux kernels with a reproducing property and an ABC-type identity. The second half studies diagonal Christoffel and Geronimus perturbations of the moment matrix and derives quasi-determinantal Christoffel formulas for the perturbed polynomials. The central structural results follow by explicit linear algebra from the stated factorization, but the Geronimus section with singular part contains an internal inconsistency: the singular part is supported off the unit circle and hence does not contribute to the moment matrix, yet the Christoffel–Geronimus formulas depend on its masses.","tokens_in":54358,"tokens_out":9352,"duration_ms":92233,"significance":"If the Geronimus issue is resolved, the paper would be a useful unified treatment of multiple orthogonal Laurent polynomials on the unit circle, with explicit banded recurrences and concrete Christoffel and Geronimus formulas. The derivations are mostly transparent and are not fitted to the conclusions; the Christoffel perturbation formulas in Propositions 3.3 and 3.6 are explicit and could be checked numerically. However, the degree-bound lemmas rely on an absent appendix, and the mass-dependent Geronimus formulas are not connected to the moment matrix as defined. The paper therefore cannot be accepted in its present form.","major_comments":[{"comment":"The singular part dμ_s is supported at the zeros of W[q], which the text explicitly says are not on T. Hence ∫_T f dμ_s = 0 for every integrand f, so the moment matrix \\tilde M in Proposition 4.1 is independent of the masses m_{b,a,j}. Consequently the Gauss–Borel factors \\tilde L, \\tilde U and the polynomials \\tilde B, \\tilde A are mass-independent. Proposition 4.7 nevertheless gives a Christoffel–Geronimus formula for \\tilde A whose entries F^{(b)}_{n,j} contain the masses m_{b,a,j}, and Section 4.5 does the same for \\tilde B. Unless those mass terms cancel identically—which is neither shown nor plausible—the formula cannot describe the \\tilde A defined from \\tilde M. This is a load-bearing inconsistency, not a minor gap; the Geronimus section needs to either place the singular part on T or redefine the moment matrix and all integrals over the full support of d\\tilde μ.","section":"Section 4.2, Eq. (94), Remark after Eq. (96), Props. 4.1 and 4.7"},{"comment":"The proof of Lemma 2.7 is deferred with the words 'See Appendix,' but no appendix appears in the arXiv v1, and Lemma 2.9 is stated without proof as analogous. These degree bounds are load-bearing: Corollary 2.16 uses them to write the mixed orthogonality index ranges, and Proposition 4.6 uses them to locate the vanishing of the orthogonality integrals in the Geronimus argument. The authors must include the appendix or provide the proofs in the main text before the degree claims can be verified.","section":"Lemma 2.7 and Lemma 2.9"},{"comment":"The Cauchy transform \\tilde C(z) is written as an integral over T plus a functional pairing against dμ_s. If dμ_s is supported off T, the integral over T gives zero and the pairing is not the same object as the moment integral used in Proposition 4.1. The text should define rigorously whether all moment-type integrals are over T or over the full support of d\\tilde μ; the current mixed convention is the source of the inconsistency flagged above.","section":"Section 4.2, Eq. (96)"}],"minor_comments":[{"comment":"The proof is omitted as 'straightforward'; please include a short proof or at least specify the exact truncation used, since this corollary underpins the reproducing property in Theorem 2.42.","section":"Corollary 2.14"},{"comment":"The Geronimus moment matrix is sometimes denoted \\check M and sometimes \\tilde M, and the perturbed polynomials are denoted \\check B, \\check A in (84) but \\tilde B, \\tilde A elsewhere; please unify the notation throughout Section 4.","section":"Section 4, notation"},{"comment":"There are several typos and minor infelicities, such as 'wan to' in the Acknowledgments and a dangling reference to a non-existent Appendix in Lemma 2.7; these should be corrected in a revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is a direct continuation of the authors' previous work on CMV biorthogonal Laurent polynomials and multiple orthogonal polynomials; the novelty lies in the multiple-index CMV ordering and the perturbation formulas. The main technical blocker is the Geronimus singular-part inconsistency: the mass-dependent formulas in Proposition 4.7 and Section 4.5 do not describe the polynomials obtained from the moment matrix as defined. If the authors can repair this by choosing the singular part on T or by consistently redefining the moment data, the paper could be suitable for publication. The absent appendix for the degree lemmas should also be addressed in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The good news first: the construction of mixed multiple orthogonal Laurent polynomials on the unit circle via Gauss–Borel factorization of a rectangular CMV moment matrix is genuinely new. The banded recurrences (Propositions 2.24–2.25), the Szegő matrices, the CD kernel and ABC theorem, and the diagonal Christoffel perturbation formulas are all worked out in explicit linear-algebraic detail. The degree bounds in Lemma 2.7 are plausible; the proof is deferred to an appendix that does not appear in the posted version, which is a minor completeness issue if the appendix is supplied. The Christoffel section looks coherent and sound.\n\nThe real problem is Section 4.2–4.5, the Geronimus perturbation with singular part. The singular part dμ_s is a sum of Dirac deltas placed at the zeros of W_b, which are off the unit circle. The paper itself says dμ_s is not supported on T. That means every integral defining the moment matrix ˇM = ∮_T Z dˇμ Z^T is blind to dμ_s: ˇM equals the moment matrix of W^{-1}dμ alone. Consequently the Gauss–Borel factors ˇL, ˇU, and hence ˇA and ˇB, are independent of the masses m_{b,a,j}. Yet Propositions 4.7 and 4.10 deliver Christoffel–Geronimus formulas for ˇA and ˇB whose entries F^{(b)}_{n,j} explicitly depend on the masses. Unless those mass terms cancel identically—neither shown nor plausible—these formulas cannot describe the polynomials obtained from the stated moment matrix. This is an internal inconsistency, not a missing proof or a mild assumption. It could be repaired by placing the singular support on T, but that would be a different perturbation and would require reworking the section.\n\nThe non-singular Geronimus part and the Christoffel part are not affected by this flaw. So the paper is a mixed bag: Sections 2–3 and the regular Geronimus part deserve serious refereeing; the singular-part Geronimus section needs substantial correction before the paper can be accepted as a whole.\n\nI would send this to a qualified referee, with a specific instruction to scrutinize Section 4.2–4.5 against the definition of ˇM. The paper is useful to specialists in multiple orthogonal polynomials, CMV matrices, and integrable systems. It is not ready in its current form, but the core framework is solid enough to warrant referee time.","headline":"Main CMV Gauss–Borel framework is new and mostly sound; singular-part Geronimus section has a load-bearing mass inconsistency.","tokens_in":54849,"tokens_out":6027,"would_cite":false,"duration_ms":58713,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33C45","33C47","42C05","15A23"],"pacs":[],"model":"deepseek-v4-flash","headline":"Mixed multiple orthogonal Laurent polynomials on the unit circle are built from CMV moment matrices via Gauss–Borel factorization, with banded recurrences and explicit Christoffel/Geronimus formulas as quasi-determinants.","keywords":["Mixed multiple orthogonal Laurent polynomials","unit circle","Christoffel–Darboux formulas","ABC theorem","recurrence relations","Christoffel perturbations","Geronimus perturbations","Gauss–Borel factorization"],"falsifier":"Pick the scalar case $q=p=1$ with a measure $\\mathrm d\\mu=w(\\theta)\\,\\mathrm d\\theta/(2\\pi)$ for a smooth positive weight, compute the first $N$ moments, perform the Gauss–Borel factorization, and check numerically that $T$ is pentadiagonal and $TB(z)=zB(z)$ holds for large $N$. Then take a prepared perturbation $W(z)=z^{-1}(z-z_1)(z-z_2)$ with two distinct zeros, form the $2\\times 2$ matrix $\\mathbb B_n=\\begin{pmatrix} B_n(z_1)&B_n(z_2)\\\\ B_{n+1}(z_1)&B_{n+1}(z_2)\\end{pmatrix}$, and compare the quasi-determinantal output of Proposition 3.3 with polynomials obtained by direct orthogonalization of the perturbed measure $W\\,\\mathrm d\\mu$. If $\\det \\mathbb B_n=0$ for some $n$ or the two computations disagree, the paper's central formula fails in the stated generality.","tokens_in":53771,"feed_emoji":"📐","tokens_out":12650,"duration_ms":115678,"temperature":0.7,"pith_summary":"The paper sets out to build a theory of mixed multiple orthogonal Laurent polynomials on the unit circle (MOLPUC), a many-measure analogue of the classical Szegő/CMV theory. It starts from a $q\\times p$ matrix of complex measures on the unit circle, arranges its moments in a CMV-type block matrix, and uses the Gauss–Borel factorization of that moment matrix to define two families of block Laurent polynomials $B(z)$ and $A(z^{-1})$. The paper proves that these families satisfy simultaneous biorthogonality and mixed multiple orthogonality relations, have sharp degree bounds, and obey finite-band recurrences of the form $T B(z)=z B(z)$ and $\\bar A(z^{-1})T=z\\bar A(z^{-1})$. On top of this, it derives Christoffel–Darboux kernels with a reproducing property and an ABC-type representation, and then gives explicit Christoffel formulas for diagonal Christoffel and Geronimus perturbations of the measure matrix, written as quasi-determinants. If the construction is correct, it provides a single algebraic framework that contains the scalar unit-circle theory and extends it to a genuinely multiple-measure setting.","feed_headline":"One banded matrix carries mixed orthogonal Laurent polynomials","feed_subtitle":"Gauss–Borel factorization yields recurrences and quasi-determinantal Christoffel and Geronimus formulas.","key_machinery":"The load-bearing object is the pair of semi-infinite CMV moment matrices $M=\\oint_{\\mathbb T} Z_{[q]}(z)\\,\\mathrm d\\mu(z)\\,Z_{[p]}^{\\top}(z^{-1})$ and its counterpart $\\mathcal M$, together with their Gauss–Borel factorizations $M=L^{-1}\\bar U^{-1}$ and $\\mathcal M=\\bar{\\mathcal L}^{-1}\\mathcal U^{-1}$. The CMV monomial matrix $Z_{[r]}(z)$ alternates blocks $I_r, z^{-1}I_r, zI_r, z^{-2}I_r,\\dots$; multiplying it by the sparse banded spectral matrix $\\Upsilon_{[r]}$ gives $\\Upsilon_{[r]}Z_{[r]}(z)=zZ_{[r]}(z)$. The conjugation $T=L\\Upsilon_{[q]}L^{-1}=\\bar U^{-1}\\Upsilon_{[p]}\\bar U$ converts this monomial eigenrelation into a finite-band recurrence for the actual Laurent polynomials. For perturbations, the key object is the connector (connection) matrix $N_C=\\hat L W L^{-1}=\\hat U^{-1}U$, whose upper $(2dq+1)$-banded structure is forced by the balanced form of $W$, together with the quasi-determinant (Schur complement) that encodes the Christoffel formulas.","core_discovery":"The paper's central discovery is that the Gauss–Borel factorization of the left and right CMV moment matrices $M$ and $\\mathcal M$ produces Laurent polynomial matrices $B(z)=LZ_{[q]}(z)$ and $A(z^{-1})=Z_{[p]}^{\\top}(z^{-1})U$ that are biorthogonal with respect to the matrix of measures and satisfy the banded eigenmatrix recurrences $T B(z)=z B(z)$ and $\\bar A(z^{-1})T=z\\bar A(z^{-1})$, where $T=L\\Upsilon_{[q]}L^{-1}=\\bar U^{-1}\\Upsilon_{[p]}\\bar U$ has exactly $2(p+q)+1$ diagonals (Propositions 2.24 and 2.25). The companion Szegő matrices $R,S,\\mathcal R,\\mathcal S$ factor $T$ as $T=SR$ and $T^{-1}=\\mathcal R\\mathcal S$. Using these recurrences, the paper constructs Christoffel–Darboux kernels that reproduce under the matrix measure and admit the ABC representation $K^{[n]}(x,y)=(Z_{[p]}^{\\top}(x^{-1}))^{[n]}(M^{[n]})^{-1}(Z_{[q]}(y))^{[n]}$. For a diagonal perturbation by a balanced Laurent polynomial $W$ with $2dq$ simple zeros, the perturbed polynomials $\\hat B$ and $\\hat A$ are expressed as quasi-determinants built from $B$ and $A$ evaluated at those zeros; the Geronimus analogue uses Cauchy transforms and second-kind functions. A scalar reduction recovers the classical Szegő recurrences with Verblunsky-type coefficients.","pith_inferences":["If the banded eigenmatrix $T$ exists for any matrix of measures admitting the Gauss–Borel factorization, the same mechanism should yield a spectral theorem for banded operators with positive bidiagonal factorization in the mixed multiple setting, parallel to the known one-measure result.","The quasi-determinantal Christoffel formulas suggest a practical numerical route for perturbed problems: one only needs to invert the $2dq\\times 2dq$ evaluation matrix at the zeros of $W$, rather than rerun the full orthogonalization; this computational claim goes beyond what the paper states.","The off-circle support of the Geronimus singular part in Section 4.2 indicates that the resulting object is a matrix of functionals rather than a measure on $\\mathbb T$; this may connect the theory to spectral measures of non-unitary operators or to related multi-measure systems on the circle.","The authors' remark that their orthogonality relates to a weighted variant of earlier multiple orthogonal polynomials on the unit circle suggests that a change-of-measure identity could make the prior construction a special case; verifying such an identity would be a direct testable extension."],"forward_implications":["The recurrences $TB(z)=zB(z)$ and $\\bar A(z^{-1})T=z\\bar A(z^{-1})$ are valid with $T$ a banded matrix of $2(p+q)+1$ diagonals, so the full sequence of block Laurent polynomials is governed by a finite-width recursion.","The Christoffel–Darboux kernel $K^{[n]}(x,y)$ satisfies the reproducing property under the matrix measure and admits the ABC-type inverse-moment representation $K^{[n]}(x,y)=(Z_{[p]}^{\\top}(x^{-1}))^{[n]}(M^{[n]})^{-1}(Z_{[q]}(y))^{[n]}$.","Under a diagonal Christoffel perturbation by a matrix balanced Laurent polynomial $W$, the perturbed polynomials $\\hat B$ and $\\hat A$ are expressed as quasi-determinants divided by $W_b(z)$ and by the appropriate evaluation matrix, giving explicit finite-dimensional formulas.","Under a Geronimus perturbation with singular part, the biorthogonal polynomials $\\check A$ and $\\check B$ are given by quasi-determinantal Christoffel–Geronimus formulas involving Cauchy transforms and second-kind functions.","In the scalar case $p=q=1$ with real measure, the general machinery reduces to the classical Szegő recurrences with Verblunsky-type coefficients, so the paper contains the scalar CMV theory as a special case."],"supporting_citations":[{"why":"Introduces mixed multiple orthogonal polynomials, the object class this paper extends to the unit circle.","marker":"[33]"},{"why":"Develops matrix orthogonality for vector polynomials, providing the rectangular weight-matrix formulation used throughout.","marker":"[34]"},{"why":"Introduces the CMV ordering and five-diagonal matrices for orthogonal polynomials on the unit circle, the basis for the Laurent polynomial ordering.","marker":"[15]"},{"why":"Extends the CMV ordering to orthogonal Laurent polynomials via Gauss–Borel factorization, the method adapted here.","marker":"[2]"},{"why":"Establishes the Gauss–Borel factorization approach for mixed multiple orthogonal polynomials and its connection to integrable systems.","marker":"[1]"},{"why":"Earlier construction of multiple orthogonal polynomials on the unit circle; the present orthogonality is described as a weighted variant of it.","marker":"[29]"},{"why":"Develops CMV biorthogonal Laurent polynomial perturbations and Christoffel formulas, the template for Section 3.","marker":"[6]"},{"why":"Provides matrix Geronimus transformations, the model for the Geronimus perturbation section.","marker":"[4]"},{"why":"Gives the connector-matrix and quasi-determinant (Schur complement) machinery used to derive the Christoffel and Christoffel–Geronimus formulas.","marker":"[26]"},{"why":"Supplies the definition and calculus of quasi-determinants used in the explicit Christoffel formulas.","marker":"[22]"}],"fun_headline_variants":["Banded matrix links Gauss–Borel and Christoffel–Darboux kernels","Quasi-determinantal formulas for Christoffel and Geronimus perturbations","CMV moment matrices yield banded recurrences and ABC theorem","Multiple orthogonal Laurent polynomials via Gauss–Borel factorization","Banded eigenmatrix T drives multiple orthogonal Laurent polynomials"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes the matrix of moments can be factored as a lower times an upper triangular matrix (all leading principal minors nonzero), and the perturbation results further assume that the perturbation's $2dq$ zeros are simple, distinct, and chosen so that the evaluation matrices $\\mathbb B_n$ and $\\Pi$ are nonsingular; when those determinants vanish the stated Christoffel and Christoffel–Geronimus formulas do not hold.","fun_headline_variants_meta":{"raw":{"variants":["Banded matrix links Gauss–Borel and Christoffel–Darboux kernels","Quasi-determinantal formulas for Christoffel and Geronimus perturbations","CMV moment matrices yield banded recurrences and ABC theorem","Multiple orthogonal Laurent polynomials via Gauss–Borel factorization","Banded eigenmatrix T drives multiple orthogonal Laurent polynomials"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000302,"raw_usage":{"total_tokens":1779,"prompt_tokens":1022,"completion_tokens":757,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":667}},"tokens_in":638,"tokens_out":757,"duration_ms":6321,"temperature":1.0,"reasoning_tokens":667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:14:21.255793+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick the scalar case $q=p=1$ with a measure $\\mathrm d\\mu=w(\\theta)\\,\\mathrm d\\theta/(2\\pi)$ for a smooth positive weight, compute the first $N$ moments, perform the Gauss–Borel factorization, and check numerically that $T$ is pentadiagonal and $TB(z)=zB(z)$ holds for large $N$. Then take a prepared perturbation $W(z)=z^{-1}(z-z_1)(z-z_2)$ with two distinct zeros, form the $2\\times 2$ matrix $\\mathbb B_n=\\begin{pmatrix} B_n(z_1)&B_n(z_2)\\\\ B_{n+1}(z_1)&B_{n+1}(z_2)\\end{pmatrix}$, and compare the quasi-determinantal output of Proposition 3.3 with polynomials obtained by direct orthogonalization of the perturbed measure $W\\,\\mathrm d\\mu$. If $\\det \\mathbb B_n=0$ for some $n$ or the two computations disagree, the paper's central formula fails in the stated generality.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces mixed multiple orthogonal polynomials, the object class this paper extends to the unit circle."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops matrix orthogonality for vector polynomials, providing the rectangular weight-matrix formulation used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the CMV ordering and five-diagonal matrices for orthogonal polynomials on the unit circle, the basis for the Laurent polynomial ordering."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the CMV ordering to orthogonal Laurent polynomials via Gauss–Borel factorization, the method adapted here."},{"cited_title":"Álvarez-Fernández, U","cited_arxiv_id":null,"evidence_quote":"Establishes the Gauss–Borel factorization approach for mixed multiple orthogonal polynomials and its connection to integrable systems."},{"cited_title":"Mínguez-Ceniceros, and W","cited_arxiv_id":null,"evidence_quote":"Earlier construction of multiple orthogonal polynomials on the unit circle; the present orthogonality is described as a weighted variant of it."},{"cited_title":"Ariznabarreta, M","cited_arxiv_id":null,"evidence_quote":"Develops CMV biorthogonal Laurent polynomial perturbations and Christoffel formulas, the template for Section 3."},{"cited_title":"Ariznabarreta, J","cited_arxiv_id":null,"evidence_quote":"Provides matrix Geronimus transformations, the model for the Geronimus perturbation section."},{"cited_title":"Mañas, Revisiting biorthogonal polynomials: An LU factorization discussion","cited_arxiv_id":null,"evidence_quote":"Gives the connector-matrix and quasi-determinant (Schur complement) machinery used to derive the Christoffel and Christoffel–Geronimus formulas."},{"cited_title":"Gelfand, S","cited_arxiv_id":null,"evidence_quote":"Supplies the definition and calculus of quasi-determinants used in the explicit Christoffel formulas."}],"review_version":1}