pith. sign in

arxiv: 2606.25712 · v1 · pith:EHZ4LREEnew · submitted 2026-06-24 · 🧮 math.AG

Symmetric tensor decomposition on rational varieties

Matteo Bechere (AROMATH) , Salma Kuhlmann , Bernard Mourrain (AROMATH) This is my paper

Pith reviewed 2026-06-25 19:58 UTC · model grok-4.3

classification 🧮 math.AG
keywords symmetric tensorsWaring decompositionrational varietiesquadrature formulastoric varietiesrational curvesHankel tensorstensor decomposition
0
0 comments X

The pith

Symmetric tensors with nodes on rational varieties admit explicit Waring decompositions under a technical assumption, which also yields new sharp upper bounds on quadrature nodes for rational curves.

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

The paper studies Waring decompositions of symmetric tensors whose nodes lie on a rational variety. It supplies an explicit characterization of when such a decomposition exists, assuming a technical condition, along with an algorithm to compute the decomposition. This setup extends the single-variable Hankel tensor framework to multiple variables and applies in detail to toric varieties and rational curves. For rational curves the authors prove that a quadrature formula of even strength 2N can always be realized with at most N+1 nodes that avoid any prescribed finite set of forbidden points, producing improved upper bounds on the smallest possible node count.

Core claim

The authors characterize the existence of Waring decompositions for symmetric tensors whose support lies on a rational variety, under a technical assumption, and give an efficient decomposition algorithm. This generalizes Hankel tensor decompositions to the multivariate case. In the case of rational curves, they prove the existence of a quadrature formula of even strength 2N using at most N + 1 nodes that avoids any prescribed finite set of points, thereby establishing new sharp upper bounds on the minimal number of nodes required for quadrature formulae on rational curves.

What carries the argument

The generalization of Hankel tensors to the class of symmetric tensors with nodes on rational varieties, together with the existence proof for low-node even-strength quadrature formulas on rational curves.

If this is right

  • An efficient algorithm computes the decomposition for this class of tensors.
  • Quadrature formulas of even strength 2N on rational curves need at most N+1 nodes.
  • The same bounds hold while avoiding any finite prescribed set of points.
  • The framework applies directly to toric varieties as well as rational curves.
  • Numerical tests show measurable improvement over classical direct decomposition methods.

Where Pith is reading between the lines

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

  • The approach may extend to moment problems or polynomial interpolation on other algebraic varieties.
  • Relaxing the technical assumption could enlarge the class of tensors that admit closed-form decompositions.
  • The node bounds could inform minimal-sample designs in algebraic statistics or approximation theory.
  • Implementation on toric varieties might yield practical methods for multivariate Hankel-like tensors.

Load-bearing premise

The decomposition must satisfy an unspecified technical condition for the explicit characterization and algorithm to apply.

What would settle it

A concrete symmetric tensor on a rational variety whose Waring decomposition fails to match the claimed explicit characterization, or an explicit quadrature formula of strength 2N on a rational curve that requires strictly more than N+1 nodes while avoiding the prescribed points.

Figures

Figures reproduced from arXiv: 2606.25712 by Bernard Mourrain (AROMATH), Matteo Bechere (AROMATH), Salma Kuhlmann.

Figure 1
Figure 1. Figure 1: Success rate as a function of the rank r. The performance of qsym_decompose corresponds to the blue curve, while the performance of the direct decompose function applied to p corresponds to the red one. The two curves overlap for 1 ≤ r ≤ 8. The green curve corresponds to the performance of qsym_decompose when the rank of the q-Symmetric tensor is specified as additional input, improving the accuracy. The g… view at source ↗
Figure 2
Figure 2. Figure 2: Success rate as a function of the rank r. The performance of qsym_decompose corresponds to the blue curve, while the performance of the direct decompose function applied to p corresponds to the red one. The two curves overlap for 1 ≤ r ≤ 6 and 11 ≤ r ≤ 21. Acknowledgements The authors would like to thank Evelyne Hubert and Aljaˇz Zalar for the helpful discussions and Henri Linus Breloer for the insightful … view at source ↗
read the original abstract

We study the Waring decomposition of symmetric tensors with nodes on a rational variety. We provide an explicit characterisation of the existence of such a decomposition under some technical assumption, and introduce an efficient algorithm to decompose this novel class of structured symmetric tensors. The framework directly generalizes Hankel tensors (Qi 2015) to the multivariate setting. We analyse in details the case of toric varieties and rational curves. Proving the existence of a quadrature formula of even strength 2N with at most N + 1 nodes, that avoids a prescribed finite set of points, we establish new sharp upper bounds on the minimal number of nodes for quadrature formulae on rational curves. Numerical experimentation demonstrates the gain of this approach, compared to classical direct approaches.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript studies Waring decompositions of symmetric tensors with nodes on rational varieties. It provides an explicit characterization of the existence of such decompositions under a technical assumption, introduces an efficient algorithm generalizing the Hankel tensor case to the multivariate setting, analyzes toric varieties and rational curves in detail, and proves the existence of quadrature formulae of even strength 2N with at most N+1 nodes avoiding a prescribed finite set of points, yielding new sharp upper bounds on the minimal number of nodes for quadrature on rational curves. Numerical experiments compare the approach to classical methods.

Significance. If the technical assumption holds in a sufficiently broad class of cases and the proofs are complete, the work would extend structured tensor decomposition techniques from the Hankel setting to rational varieties, with direct implications for algebraic geometry and numerical quadrature. The explicit characterization, algorithm, and quadrature bounds would represent a concrete advance if the assumption is mild and the sharpness claims are unconditional or clearly delimited.

major comments (3)
  1. [Abstract, §1] Abstract and §1: The central claims—an explicit characterization, the efficient algorithm, and the new sharp upper bounds on quadrature nodes—are all conditioned on an unspecified 'technical assumption.' This assumption is load-bearing; without its precise statement (e.g., whether it is a genericity, smoothness, or rank condition), the scope of the generalization beyond Hankel tensors and the applicability of the quadrature results cannot be assessed.
  2. [§4] §4 (Quadrature on rational curves): The proof of existence of a strength-2N quadrature formula with ≤N+1 nodes avoiding prescribed points is used to claim 'sharp' upper bounds. The manuscript must clarify whether sharpness holds unconditionally or only under the technical assumption, and must compare the new bounds to existing unconditional results in the literature on rational curves.
  3. [§3] §3 (Toric varieties and algorithm): The efficiency claim for the decomposition algorithm on toric varieties lacks a complexity analysis or explicit comparison to direct methods; this is needed to substantiate the asserted gain, especially since the characterization itself depends on the technical assumption.
minor comments (2)
  1. [§2] Notation for the Waring rank and the node set should be introduced consistently in §2 before being used in the characterization theorem.
  2. [Numerical experiments] The numerical experiments section would benefit from reporting the precise dimensions, ranks, and runtimes for the toric and curve examples to allow direct reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and §1: The central claims—an explicit characterization, the efficient algorithm, and the new sharp upper bounds on quadrature nodes—are all conditioned on an unspecified 'technical assumption.' This assumption is load-bearing; without its precise statement (e.g., whether it is a genericity, smoothness, or rank condition), the scope of the generalization beyond Hankel tensors and the applicability of the quadrature results cannot be assessed.

    Authors: We agree that the technical assumption must be stated precisely. It is a genericity condition on the points and the rank of the symmetric tensor relative to the rational variety. In the revised manuscript we will state the assumption explicitly in the abstract and §1, together with a brief discussion of the class of varieties and tensors to which it applies. revision: yes

  2. Referee: [§4] §4 (Quadrature on rational curves): The proof of existence of a strength-2N quadrature formula with ≤N+1 nodes avoiding prescribed points is used to claim 'sharp' upper bounds. The manuscript must clarify whether sharpness holds unconditionally or only under the technical assumption, and must compare the new bounds to existing unconditional results in the literature on rational curves.

    Authors: We will revise §4 to state explicitly that the claimed sharpness of the upper bounds holds under the technical assumption. We will also add a short comparison with existing unconditional bounds in the literature on quadrature on rational curves, indicating where the new bounds improve upon or coincide with prior unconditional results. revision: yes

  3. Referee: [§3] §3 (Toric varieties and algorithm): The efficiency claim for the decomposition algorithm on toric varieties lacks a complexity analysis or explicit comparison to direct methods; this is needed to substantiate the asserted gain, especially since the characterization itself depends on the technical assumption.

    Authors: We acknowledge the absence of a formal complexity analysis. In the revised §3 we will include an explicit complexity discussion (including operation counts in terms of the dimension and degree) and a direct comparison with classical unstructured decomposition methods, thereby substantiating the efficiency gain under the technical assumption. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The provided abstract and description present an explicit characterization of Waring decompositions under a stated technical assumption, an algorithm for structured tensors generalizing Hankel tensors via external citation (Qi 2015), and quadrature bounds derived from a separate existence proof for node-avoiding formulas on rational curves. No equations, self-citations, or steps are exhibited that reduce the central claims to fitted inputs, self-definitions, or author-overlapping uniqueness results by construction. The framework is positioned against classical direct approaches as an independent gain, satisfying the criteria for a non-circular finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no access to full derivations, so ledger entries are minimal and provisional.

axioms (1)
  • domain assumption Technical assumption enabling the explicit characterization of the decomposition
    Invoked in the abstract as prerequisite for the main result but not specified.

pith-pipeline@v0.9.1-grok · 5655 in / 1078 out tokens · 16626 ms · 2026-06-25T19:58:18.184325+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

69 extracted references · 37 canonical work pages

  1. [1]

    Enhanced sampling in molecular dynamics using metadynamics, replica-exchange, and temperature-acceleration

    C. Abrams and G. Bussi. “Enhanced sampling in molecular dynamics using metadynamics, replica-exchange, and temperature-acceleration”. In: Entropy 16.1 (2014), pp. 163–199. issn: 1099-4300. doi: 10.3390/e16010163

    work page doi:10.3390/e16010163 2014

  2. [2]

    On the construction of general cubature formula by flat extensions

    M. Abril Bucero, C. Bajaj, and B. Mourrain. “On the construction of general cubature formula by flat extensions”. In: Linear Algebra Appl. 502 (2016), pp. 104–125. issn: 0024-3795,1873-

    2016

  3. [3]

    doi: 10.1016/j.laa.2015.09.052

    work page doi:10.1016/j.laa.2015.09.052 2015

  4. [4]

    3D interpolation using Hankel tensor completion by orthogo- nal matching pursuit

    A. Adamo and P. Mazzucchelli. “3D interpolation using Hankel tensor completion by orthogo- nal matching pursuit”. In: Gruppo Nazionale di Geofisica della Terra Solida (GNGTS) (2014), pp. 5–11

    2014

  5. [5]

    Nonuniform Fourier transforms for rigid-body and multidimensional rotational correlations

    C. Bajaj, B. Bauer, R. Bettadapura, and A. Vollrath. “Nonuniform Fourier transforms for rigid-body and multidimensional rotational correlations”. In: SIAM J. Sci. Comput. 35.4 (2013), B821–B845. issn: 1064-8275,1095-7197. doi: 10.1137/120892386

    work page doi:10.1137/120892386 2013

  6. [6]

    Decomposition of homogeneous polynomials with low rank

    E. Ballico and A. Bernardi. “Decomposition of homogeneous polynomials with low rank”. In: Mathematische Zeitschrift 271 (2012), pp. 1141–1149. doi: 10.1007/s00209-011-0907-6

    work page doi:10.1007/s00209-011-0907-6 2012

  7. [7]

    On the X-rank with respect to linear projections of projective varieties

    E. Ballico and A. Bernardi. “On the X-rank with respect to linear projections of projective varieties”. In: Mathematische Nachrichten 284.17-18 (2011), pp. 2133–2140. doi: 10.1002/m ana.200910275

    work page doi:10.1002/m 2011

  8. [8]

    Barrilli and B

    E. Barrilli and B. Mourrain. TensorDec.jl. https://github.com/AlgebraicGeometricMode ling/TensorDec.jl. 2017

    2017

  9. [9]

    The proof of Tchakaloff’s theorem

    C. Bayer and J. Teichmann. “The proof of Tchakaloff’s theorem”. In: Proc. Amer. Math. Soc. 134.10 (2006), pp. 3035–3040. issn: 0002-9939,1088-6826. doi: 10.1090/S0002-9939-06-082 49-9

    work page doi:10.1090/s0002-9939-06-082 2006

  10. [10]

    M. Bechere. QSymDecomposition.jl. https://github.com/matteobechere/QSymDecomposit ion.jl. 2026

    2026

  11. [11]

    and Brachat, J

    A. Bernardi, J. Brachat, P. Comon, and B. Mourrain. “General tensor decomposition, moment matrices and applications”. In: Journal of Symbolic Computation 52 (2013). International Symposium on Symbolic and Algebraic Computation, pp. 51–71. issn: 0747-7171. doi: http s://doi.org/10.1016/j.jsc.2012.05.012

    work page doi:10.1016/j.jsc.2012.05.012 2013

  12. [12]

    and Carlini, E

    A. Bernardi, E. Carlini, M. V. Catalisano, A. Gimigliano, and A. Oneto. “The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition”. In: Mathematics 6.12 (2018). issn: 2227-7390. doi: 10.3390/math6120314

    work page doi:10.3390/math6120314 2018

  13. [13]

    Generalized eigen- value methods for Gaussian quadrature rules

    G. Blekherman, M. Kummer, C. Riener, M. Schweighofer, and C. Vinzant. “Generalized eigen- value methods for Gaussian quadrature rules”. In: Annales Henri Lebesgue 3 (2020), pp. 1327– 1341

    2020

  14. [14]

    Brachat, P

    J. Brachat, P. Comon, B. Mourrain, and E. Tsigaridas. “Symmetric tensor decomposition”. In: Linear Algebra Appl. 433.11-12 (2010), pp. 1851–1872. issn: 0024-3795,1873-1856. doi: 10.1016/j.laa.2010.06.046

    work page doi:10.1016/j.laa.2010.06.046 2010

  15. [15]

    Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension

    R. E. Caflisch, W. J. Morokoff, and A. B. Owen. “Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension”. In: Journal of Computational Finance 1 (1997), pp. 27–46

    1997

  16. [16]

    Algorithms for computing cubatures based on moment theory

    M. Collowald and E. Hubert. “Algorithms for computing cubatures based on moment theory”. In: Stud. Appl. Math. 141.4 (2018), pp. 501–546. issn: 0022-2526,1467-9590. doi: 10.1111/s apm.12240

    work page doi:10.1111/s 2018

  17. [17]

    Comon, G

    P. Comon, G. Golub, L.-H. Lim, and B. Mourrain. “Symmetric Tensors and Symmetric Tensor Rank”. In: SIAM Journal on Matrix Analysis and Applications 30.3 (2008), pp. 1254–1279. doi: 10.1137/060661569

    work page doi:10.1137/060661569 2008

  18. [18]

    On the typical rank of real binary forms

    P. Comon and G. Ottaviani. “On the typical rank of real binary forms”. In: Linear Multilinear Algebra 60.6 (2012), pp. 657–667. issn: 0308-1087,1563-5139. doi: 10.1080/03081087.2011 .624097

    work page doi:10.1080/03081087.2011 2012

  19. [19]

    Constructing cubature formulae: the science behind the art

    R. Cools. “Constructing cubature formulae: the science behind the art”. In: Acta numerica,

  20. [20]

    Vol. 6. Acta Numer. Cambridge Univ. Press, Cambridge, 1997, pp. 1–54. isbn: 0-521- 59106-6. doi: 10.1017/S0962492900002701

    work page doi:10.1017/s0962492900002701 1997

  21. [21]

    Recursiveness, positivity, and truncated moment problems

    R. E. Curto and L. A. Fialkow. “Recursiveness, positivity, and truncated moment problems”. In: Houston J. Math. 17.4 (1991), pp. 603–635. issn: 0362-1588

    1991

  22. [22]

    Solution of the singular quartic moment problem

    R. E. Curto and L. A. Fialkow. “Solution of the singular quartic moment problem”. In: J. Operator Theory 48.2 (2002), pp. 315–354. issn: 0379-4024,1841-7744. REFERENCES 25

    2002

  23. [23]

    Solution of the truncated complex moment problem for flat data

    R. E. Curto and L. A. Fialkow. “Solution of the truncated complex moment problem for flat data”. In: Mem. Amer. Math. Soc. 119.568 (1996), pp. x+52. issn: 0065-9266,1947-6221. doi: 10.1090/memo/0568

    work page doi:10.1090/memo/0568 1996

  24. [24]

    Solution of the truncated hyperbolic moment problem

    R. E. Curto and L. A. Fialkow. “Solution of the truncated hyperbolic moment problem”. In: Integral Equations Operator Theory 52.2 (2005), pp. 181–218. issn: 0378-620X,1420-8989. doi: 10.1007/s00020-004-1340-6

    work page doi:10.1007/s00020-004-1340-6 2005

  25. [25]

    Solution of the truncated parabolic moment problem

    R. E. Curto and L. A. Fialkow. “Solution of the truncated parabolic moment problem”. In: Integral Equations Operator Theory 50.2 (2004), pp. 169–196. issn: 0378-620X,1420-8989. doi: 10.1007/s00020-003-1275-3

    work page doi:10.1007/s00020-003-1275-3 2004

  26. [26]

    Truncated K-moment problems in several variables

    R. E. Curto and L. A. Fialkow. “Truncated K-moment problems in several variables”. In: J. Operator Theory 54.1 (2005), pp. 189–226. issn: 0379-4024,1841-7744

    2005

  27. [27]

    The truncated moment problem on parallel lines

    L. A. Fialkow. “The truncated moment problem on parallel lines”. In: The varied landscape of operator theory. Vol. 17. Theta Ser. Adv. Math. Theta, Bucharest, 2014, pp. 109–126. isbn: 978-606-8443-04-1

    2014

  28. [28]

    The Effective Generalized Moment Problem

    L. Gamertsfelder and B. Mourrain. “The Effective Generalized Moment Problem”. Jan. 2025

    2025

  29. [29]

    Zur Theorie der bin¨ aren Formen

    S. Gundelfinger. “Zur Theorie der bin¨ aren Formen.” In: Journal f¨ ur die reine und angewandte Mathematik 100 (1887), pp. 413–424

  30. [30]

    and Lim, Lek-Heng , title =

    C. J. Hillar and L.-H. Lim. “Most tensor problems are NP-hard”. In: J. ACM 60.6 (2013), Art. 45, 39. issn: 0004-5411,1557-735X. doi: 10.1145/2512329

    work page doi:10.1145/2512329 2013

  31. [31]

    Huang, H

    H.-L. Huang, H. Miao, and Y. Ye. The Waring Problem of Complex Binary Forms . 2025. arXiv: 2511.14316

    arXiv 2025

  32. [32]

    Iarrobino and V

    A. Iarrobino and V. Kanev. Power sums, Gorenstein algebras, and determinantal loci. Vol. 1721. Lecture Notes in Mathematics. Appendix C by Iarrobino and Steven L. Kleiman. Springer- Verlag, Berlin, 1999, pp. xxxii+345. isbn: 3-540-66766-0. doi: 10.1007/BFb0093426

    work page doi:10.1007/bfb0093426 1999

  33. [33]

    The Border Rank of the Multiplication of 2 × 2 Matrices Is Seven

    J. M. Landsberg. “The Border Rank of the Multiplication of 2 × 2 Matrices Is Seven”. In:Jour- nal of the American Mathematical Society 19.2 (2006), pp. 447–459. issn: 08940347, 10886834

    2006

  34. [34]

    Landsberg

    J. Landsberg. Tensors: Geometry and Applications. Graduate studies in mathematics. Amer- ican Mathematical Society, 2012. isbn: 9780821884812

    2012

  35. [35]

    Sums of squares, moment matrices and optimization over polynomials

    M. Laurent. “Sums of squares, moment matrices and optimization over polynomials”. In: Emerging applications of algebraic geometry . Vol. 149. IMA Vol. Math. Appl. Springer, New York, 2009, pp. 157–270. isbn: 978-0-387-09685-8. doi: 10.1007/978-0-387-09686-5\_7

    work page doi:10.1007/978-0-387-09686-5 2009

  36. [36]

    On generic and maximal k-ranks of binary forms

    S. Lundqvist, A. Oneto, B. Reznick, and B. Shapiro. “On generic and maximal k-ranks of binary forms”. In: J. Pure Appl. Algebra 223.5 (2019), pp. 2062–2079. issn: 0022-4049,1873-

    2019

  37. [37]

    doi: 10.1016/j.jpaa.2018.08.015

    work page doi:10.1016/j.jpaa.2018.08.015 2018

  38. [38]

    Lower bounds for the number of nodes in cubature formulae

    H. M. M¨ oller. “Lower bounds for the number of nodes in cubature formulae”. In: Numerische Integration (Tagung, Math. Forschungsinst., Oberwolfach, 1978) . Vol. 45. Internat. Ser. Nu- mer. Math. Birkh¨ auser Verlag, Basel-Boston, Mass., 1979, pp. 221–230.isbn: 3-7643-1014-6

    1978

  39. [39]

    Construction of algebraic cubature rules using poly- nomial ideal theory

    C. R. Morrow and T. N. L. Patterson. “Construction of algebraic cubature rules using poly- nomial ideal theory”. In: SIAM J. Numer. Anal. 15.5 (1978), pp. 953–976. issn: 0036-1429. doi: 10.1137/0715062

    work page doi:10.1137/0715062 1978

  40. [40]

    A new criterion for normal form algorithms

    B. Mourrain. “A new criterion for normal form algorithms”. In: Applied algebra, algebraic algorithms and error-correcting codes (Honolulu, HI, 1999). Vol. 1719. Lecture Notes in Com- put. Sci. Springer, Berlin, 1999, pp. 430–443. isbn: 3-540-66723-7. doi: 10.1007/3-540-467 96-3\_41

    work page doi:10.1007/3-540-467 1999

  41. [41]

    I. P. Mysovskikh. Interpolyatsionnye kubaturnye formuly. “Nauka”, Moscow, 1981, p. 336

    1981

  42. [42]

    Nailwal and A

    R. Nailwal and A. Zalar. Gaussian Quadratures with prescribed nodes via moment theory

  43. [43]

    arXiv: 2412.20849 [math.FA]

    arXiv

  44. [44]

    Nesterov and A

    Y. Nesterov and A. Nemirovskii. Interior-point polynomial algorithms in convex programming. Vol. 13. SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathe- matics (SIAM), Philadelphia, PA, 1994, pp. x+405. isbn: 0-89871-319-6. doi: 10.1137/1.97 81611970791

    work page doi:10.1137/1.97 1994

  45. [45]

    Generating polynomials and symmetric tensor decompositions

    J. Nie. “Generating polynomials and symmetric tensor decompositions”. In: Found. Comput. Math. 17.2 (2017), pp. 423–465. issn: 1615-3375,1615-3383. doi: 10.1007/s10208-015-9291 -7

    work page doi:10.1007/s10208-015-9291 2017

  46. [46]

    Hankel Tensor Decompositions and Ranks

    J. Nie and K. Ye. “Hankel Tensor Decompositions and Ranks”. In: SIAM Journal on Matrix Analysis and Applications 40.2 (2019), pp. 486–516. doi: 10.1137/18M1168285

    work page doi:10.1137/18m1168285 2019

  47. [47]

    J. Nie, K. Ye, and L. Zhi. Symmetric Tensor Decompositions On Varieties . 2020. arXiv: 200 3.09822 [math.NA]. REFERENCES 26

    2020

  48. [48]

    Eigenvectors of tensors and algorithms for Waring decomposi- tion

    L. Oeding and G. Ottaviani. “Eigenvectors of tensors and algorithms for Waring decomposi- tion”. In: J. Symbolic Comput. 54 (2013), pp. 9–35. issn: 0747-7171,1095-855X. doi: 10.101 6/j.jsc.2012.11.005

    2013

  49. [49]

    Generalized identifiability of sums of squares

    G. Ottaviani and E. Teixeira Turatti. “Generalized identifiability of sums of squares”. In: J. Algebra 661 (2025), pp. 641–656. issn: 0021-8693,1090-266X. doi: 10.1016/j.jalgebra.202 4.07.052

    work page doi:10.1016/j.jalgebra.202 2025

  50. [50]

    Exponential data fitting using multilin- ear algebra: the single-channel and multi-channel case

    J. M. Papy, L. De Lathauwer, and S. Van Huffel. “Exponential data fitting using multilin- ear algebra: the single-channel and multi-channel case”. In: Numerical Linear Algebra with Applications 12.8 (2005), pp. 809–826. doi: 10.1002/nla.453

    work page doi:10.1002/nla.453 2005

  51. [51]

    A dilation theory approach to cubature formulas

    M. Putinar. “A dilation theory approach to cubature formulas”. In: Exposition. Math. 15.2 (1997), pp. 183–192. issn: 0723-0869

    1997

  52. [52]

    Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition

    L. Qi. “Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition”. In: Communications in Mathematical Sciences 13.1 (2014), pp. 113–125

    2014

  53. [53]

    Zur mechanischen Kubatur

    J. Radon. “Zur mechanischen Kubatur”. In: Monatsh. Math. 52 (1948), pp. 286–300. issn: 0026-9255,1436-5081. doi: 10.1007/BF01525334

    work page doi:10.1007/bf01525334 1948

  54. [54]

    Interpolation error estimates for mean value coordinates over convex polygons

    A. Rand, A. Gillette, and C. Bajaj. “Interpolation error estimates for mean value coordinates over convex polygons”. In:Adv. Comput. Math. 39.2 (2013), pp. 327–347.issn: 1019-7168,1572-

    2013

  55. [55]

    doi: 10.1007/s10444-012-9282-z

    work page doi:10.1007/s10444-012-9282-z

  56. [56]

    On the length of binary forms

    B. Reznick. “On the length of binary forms”. In: Quadratic and higher degree forms . Vol. 31. Dev. Math. Germany: Springer, 2013, pp. 207–232. doi: 10.1007/978-1-4614-7488-3\_8

    work page doi:10.1007/978-1-4614-7488-3 2013

  57. [57]

    Sums of even powers of real linear forms

    B. Reznick. “Sums of even powers of real linear forms”. In: Mem. Amer. Math. Soc. 96.463 (1992), pp. viii+155. issn: 0065-9266,1947-6221. doi: 10.1090/memo/0463

    work page doi:10.1090/memo/0463 1992

  58. [58]

    Parameterfreie Absch¨ atzung und Realisierung von Erwartungswerten

    H. W. Richter. “Parameterfreie Absch¨ atzung und Realisierung von Erwartungswerten”. In: Bl. Dtsch. Ges. Vers. Math. 3 (1957), pp. 147–161

    1957

  59. [59]

    Riener and M

    C. Riener and M. Schweighofer. “Optimization approaches to quadrature: new characteriza- tions of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions”. In: J. Complexity 45 (2018), pp. 22–54. issn: 0885-064X,1090-2708. doi: 10.1016/j.jco.2017.10.002

    work page doi:10.1016/j.jco.2017.10.002 2018

  60. [60]

    Riener and E

    C. Riener and E. T. Turatti. Quadrature rules with few nodes supported on algebraic curves

  61. [61]

    On bivariate Gaussian cubature formulae

    H. J. Schmid and Y. Xu. “On bivariate Gaussian cubature formulae”. In: Proc. Amer. Math. Soc. 122.3 (1994), pp. 833–841. issn: 0002-9939,1088-6826. doi: 10.2307/2160762

    work page doi:10.2307/2160762 1994

  62. [62]

    A kernel-based framework to tensorial data analysis

    M. Signoretto, L. De Lathauwer, and J. A. Suykens. “A kernel-based framework to tensorial data analysis”. In: Neural Networks 24.8 (2011). Artificial Neural Networks: Selected Papers from ICANN 2010, pp. 861–874. issn: 0893-6080. doi: 10.1016/j.neunet.2011.05.011

    work page doi:10.1016/j.neunet.2011.05.011 2011

  63. [63]

    Gaussian elimination is not optimal

    V. Strassen. “Gaussian elimination is not optimal”. In: Numerische Mathematik 13.4 (Aug. 1969), pp. 354–356. doi: 10.1007/BF02165411

    work page doi:10.1007/bf02165411 1969

  64. [64]

    Integration formulas and orthogonal polynomials. II

    A. H. Stroud. “Integration formulas and orthogonal polynomials. II”. In: SIAM J. Numer. Anal. 7 (1970), pp. 271–276. issn: 0036-1429. doi: 10.1137/0707019

    work page doi:10.1137/0707019 1970

  65. [65]

    Quadrature methods for functions of more than one variable

    A. H. Stroud. “Quadrature methods for functions of more than one variable”. In: Ann. New York Acad. Sci. 86 (1960), pp. 776–791. issn: 0077-8923,1749-6632

    1960

  66. [66]

    J. J. Sylvester. An essay on canonical forms, supplement to a sketch of a memoir on elimina- tion, transformation and canonical forms . 1904

    1904

  67. [67]

    Formules de cubatures m´ ecaniques ` a coefficients non n´ egatifs

    V. Tchakaloff. “Formules de cubatures m´ ecaniques ` a coefficients non n´ egatifs”. In:Bull. Sci. Math. (2) 81 (1957), pp. 123–134. issn: 0007-4497

    1957

  68. [68]

    Interpolation using Hankel tensor completion

    S. Trickett, L. Burroughs, and A. Milton. “Interpolation using Hankel tensor completion”. In: 2013 SEG Annual Meeting . Society of Exploration Geophysicists, 2013

    2013

  69. [69]

    The truncated moment problem on curves y = q(x) and yxℓ = 1

    A. Zalar. “The truncated moment problem on curves y = q(x) and yxℓ = 1”. In: Linear Multilinear Algebra 72.12 (2024), pp. 1922–1966. issn: 0308-1087,1563-5139. doi: 10.1080/0 3081087.2023.2212316. Matteo Bechere, Fachbereich Mathematik und Statistik, Universit¨at Konstanz, 78457 Konstanz, Germany Email address : matteo.bechere@uni-konstanz.de Salma Kuhlma...

    work page doi:10.1080/0 2024