In this paper, we develop a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring, in particular in any characteristic. This formula also provides a new method for evaluating and computing this discriminant efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we obtain new properties and computational rules such as the covariance and the invariance formulas. Finally, we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. Actually, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.
Valtancoli in his paper entitled (P. Valtancoli, Canonical transformations and minimal length, J. Math. Phys. 56, 122107 2015) has shown how the deformation of the canonical transformations can be made compatible with the deformed Poisson brackets. Based on this work and through an appropriate canonical transformation, we solve the problem of one dimensional (1D) damped harmonic oscillator at the classical limit of the Snyder-de Sitter (SdS) space. We show that the equations of the motion can be described by trigonometric functions with frequency and period depending on the deformed and the damped parameters. We eventually discuss the influences of these parameters on the motion of the system.
In this paper, we study the action of the rational quantum Calogero-Moser system on polynomials. In this vein, we study polynomials ring over the complex field ℂ as a module over a ring of differential operators by elaborating its irreducible submodules. we endowed the polynomial ring ℂ[x
1
, …, x
n
] with a differential structure by using directly the action of the Weyl algebra associated with the ring of symmetric polynomial ℂ[x
1
, …, x
n
]
Sn after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the symmetric group. We use the representation theory of symmetric groups to exhibit the generators of its simple components.
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