Numerically Testing Generically Reduced Projective Schemes for the Arithmetic Gorenstein Property

Daleo, Noah S.; Hauenstein, Jonathan D.

doi:10.1007/978-3-319-32859-1_11

Cited by 4 publications

(8 citation statements)

References 10 publications

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“…We used the approach in [, § 2] with Bertini to compute a so‐called pseudowitness set for

X

yielding

deg X = 280

. With this pseudowitness set, shows that

X

is arithmetically Cohen‐Macaulay and arithmetically Gorenstein. In particular, the Hilbert function of the finite set

X \cap L

where

L \subset {double-struckP}^{55}

is a general linear space of dimension 2 is

\begin{matrix} right 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 65, 75, 85, 95, 105, 115, 125, 135, 145, 155, 165, 175, \\ right 185, 195, 205, 215, 225, 235, 244, 252, 259, 265, 270, 274, 277, 279, 280, 280 . \end{matrix}

Thus, the ideal of

X \cap L

is minimally generated by a degree 10 polynomial (corresponding to

T_{10}

) and a polynomial of degree 28.…”

Section: The Polynomials Sm⟨n⟩s and Sm⟨n⟩s0mentioning

confidence: 99%

Polynomials and the exponent of matrix multiplication

Chiantini

Hauenstein

Ikenmeyer

et al. 2018

Bull. London Math. Soc.

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The exponent of matrix multiplication is the smallest constant ω such that two n × n matrices may be multiplied by performing O(n ω+ ) arithmetic operations for every > 0. Determining the constant ω is a central question in both computer science and mathematics. Strassen [Linear Algebra Appl. 52/53 (1983) 645-685] showed that ω is also governed by the tensor rank of the matrix multiplication tensor. We define certain symmetric tensors, that is, cubic polynomials, and our main result is that their symmetric rank also grows with the same exponent ω, so that ω can be computed in the symmetric setting, where it may be easier to determine. In particular, we study the symmetrized matrix multiplication tensor sM n defined on an n × n matrix A by sM n (A) = trace(A 3 ). The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent ω.

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“…We used the approach in [, § 2] with Bertini to compute a so‐called pseudowitness set for

X

yielding

deg X = 280

. With this pseudowitness set, shows that

X

is arithmetically Cohen‐Macaulay and arithmetically Gorenstein. In particular, the Hilbert function of the finite set

X \cap L

where

L \subset {double-struckP}^{55}

is a general linear space of dimension 2 is

\begin{matrix} right 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 65, 75, 85, 95, 105, 115, 125, 135, 145, 155, 165, 175, \\ right 185, 195, 205, 215, 225, 235, 244, 252, 259, 265, 270, 274, 277, 279, 280, 280 . \end{matrix}

Thus, the ideal of

X \cap L

is minimally generated by a degree 10 polynomial (corresponding to

T_{10}

) and a polynomial of degree 28.…”

Section: The Polynomials Sm⟨n⟩s and Sm⟨n⟩s0mentioning

confidence: 99%

Polynomials and the exponent of matrix multiplication

Chiantini

Hauenstein

Ikenmeyer

et al. 2018

Bull. London Math. Soc.

Self Cite

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show abstract

“…Amongst the changes from the ISSAC version of the paper (Berthomieu et al (2015)), we now mention that only Gorenstein ideals can be recovered as ideal of relations. This gives another probabilistic test for the Gorenstein property, see also Daleo and Hauenstein (2015). The Adaptive Scalar-FGLM algorithm, presented in the ISSAC paper and in Section 5, could fail on tables satisfying a relation for a while and then switching not to satisfy it anymore.…”

Section: Contributionsmentioning

confidence: 99%

“…If I = J, then J is Gorenstein. We refer to Daleo and Hauenstein (2015) for another test on the Gorenstein property of an ideal.…”

Section: Gorenstein Idealsmentioning

confidence: 99%

See 1 more Smart Citation

Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences

Berthomieu¹,

Boyer²,

Faugère³

2017

Journal of Symbolic Computation

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The so-called Berlekamp-Massey-Sakata algorithm computes a Gröbner basis of a 0-dimensional ideal of relations satisfied by an input table. It extends the Berlekamp-Massey algorithm to n-dimensional tables, for n > 1. We investigate this problem and design several algorithms for computing such a Gröbner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations. As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This FGLM-like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a multi-Hankel matrix (a multivariate generalization of Hankel matrices). Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the geometry of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode n-cyclic codes, a generalization in dimension n of Reed Solomon codes. We show that the multi-Hankel matrices are heavily structured when using the LEX ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table.

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“…Nonetheless, we note that the secant variety σ 5 (C 3 × C 4 × C 4 ) which has dimension 44, i.e., it is defective, and degree 1716. In particular, the methods of [42,43] show that σ 5 (C 3 × C 4 × C 4 ) is arithmetically Gorenstein and generated by 144 polynomials of degree 11.…”

Section: Numbermentioning

confidence: 99%

Tensor decomposition and homotopy continuation

Bernardi

Daleo

Hauenstein

et al. 2017

Differential Geometry and its Applications

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A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties X1, . . . , X k ⊂ P N defined over C. After computing ranks over C, we also explore computing real ranks. A variety of examples are included to demonstrate the numerical algebraic geometric approaches.

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Numerically Testing Generically Reduced Projective Schemes for the Arithmetic Gorenstein Property

Cited by 4 publications

References 10 publications

Polynomials and the exponent of matrix multiplication

Polynomials and the exponent of matrix multiplication

Linear algebra for computing Gröbner bases of linear recursive multidimensional sequences

Tensor decomposition and homotopy continuation

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