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Displaying 1 – 12 of 12

Testing flatness and computing rank of a module using syzygies

Oswaldo Lezama (2009)

Colloquium Mathematicae

Using syzygies computed via Gröbner bases techniques, we present algorithms for testing some homological properties for submodules of the free module $A^{m}$ , where A = R[x₁,...,xₙ] and R is a Noetherian commutative ring. We will test if a given submodule M of $A^{m}$ is flat. We will also check if M is locally free of constant dimension. Moreover, we present an algorithm that computes the rank of a flat submodule M of $A^{m}$ and also an algorithm that computes the projective dimension of an arbitrary submodule...

Testing the logarithmic comparison theorem for free divisors.

Castro-Jiménez, F.J., Ucha-Enríquez, J.M. (2004)

Experimental Mathematics

The Cayley–Menger determinant is irreducible for $n \geq 3$ .

D'Andrea, Carlos, Sombra, Martín (2005)

Sibirskij Matematicheskij Zhurnal

The cluster basis of $ℤ [x_{1, 1}, \dots, x_{3, 3}]$ .

Skandera, Mark (2007)

The Electronic Journal of Combinatorics [electronic only]

The cohomology ring of polygon spaces

Jean-Claude Hausmann, Allen Knutson (1998)

Annales de l'institut Fourier

We compute the integer cohomology rings of the “polygon spaces”introduced in [F. Kirwan, Cohomology rings of moduli spaces of vector bundles over Riemann surfaces, J. Amer. Math. Soc., 5 (1992), 853-906] and [M. Kapovich & J. Millson, the symplectic geometry of polygons in Euclidean space, J. of Diff. Geometry, 44 (1996), 479-513]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory...

The construction of orthonormal wavelets using symbolic methods and a matrix analytical approach for wavelets on the interval.

Chyzak, Frédéric, Paule, Peter, Scherzer, Otmar, Schoisswohl, Armin, Zimmermann, Burkhard (2001)

Experimental Mathematics

The F4-algorithm for Euclidean rings

Afshan Sadiq (2010)

Open Mathematics

In this short note, we extend Faugére’s F4-algorithm for computing Gröbner bases to polynomial rings with coefficients in an Euclidean ring. Instead of successively reducing single S-polynomials as in Buchberger’s algorithm, the F4-algorithm is based on the simultaneous reduction of several polynomials.

The optimal lower bound for generators of invariant rings without finite SAGBI bases with respect to any admissible order.

Göbel, Manfred (1999)

Discrete Mathematics and Theoretical Computer Science. DMTCS [electronic only]

The strong persistence property and symbolic strong persistence property

Mehrdad Nasernejad, Kazem Khashyarmanesh, Leslie G. Roberts, Jonathan Toledo (2022)

Czechoslovak Mathematical Journal

Let $I$ be an ideal in a commutative Noetherian ring $R$ . Then the ideal $I$ has the strong persistence property if and only if $(I^{k + 1} :_{R} I) = I^{k}$ for all $k$ , and $I$ has the symbolic strong persistence property if and only if $(I^{(k + 1)} :_{R} I^{(1)}) = I^{(k)}$ for all $k$ , where $I^{(k)}$ denotes the $k$ th symbolic power of $I$ . We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial ideal has the...

The tensor product of polynomials.

Schwingel, Ruth (1999)

Experimental Mathematics

Toric varieties in phylogenetics [Book]

M. Michałek (2015)

Towards the automated synthesis of a Gröbner bases algorithm.

Bruno Buchberger (2004)

RACSAM

We discuss the question of whether the central result of algorithmic Gröbner bases theory, namely the notion of S?polynomials together with the algorithm for constructing Gröbner bases using S?polynomials, can be obtained by ?artificial intelligence?, i.e. a systematic (algorithmic) algorithm synthesis method. We present the ?lazy thinking? method for theorem and algorithm invention and apply it to the ?critical pair / completion? algorithm scheme. We present a road map that demonstrates that, with...

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