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We study functions of the roots of an integer polynomial p = p (x) with m ≥ 2 distinct roots ρ = (ρ 1 , ... , ρ m) of multiplicity μ = (μ 1 , ... , μ m) , μ 1 ≥ ⋯ ≥ μ m ≥ 1. Traditionally, root functions are studied via the theory of symmetric polynomials; we generalize this theory to μ -symmetric polynomials. We initiate the study of the vector space of μ -symmetric polynomials of a given degree δ via the concepts of μ -gist and μ -ideal. In particular, we are interested in the root D μ + (r 1 , ... , r m) := ∏ 1 ≤ i < j ≤ m (r i − r j) μ i + μ j . The D-plus discriminant of p is D + (p) := D μ + (ρ 1 , ... , ρ m). This quantity appears in the complexity analysis of the root clustering algorithm of Becker et al. (ISSAC 2016). We conjecture that D μ + (r 1 , ... , r m) is μ -symmetric, which implies D + (p) is rational. To explore this conjecture experimentally, we introduce algorithms for checking if a given root function is μ -symmetric. We design three such algorithms: one based on Gröbner bases, another based on canonical bases and reduction, and the third based on solving linear equations. Each of these algorithms has variants that depend on the choice of a basis for the μ -symmetric functions. We implement these algorithms (and their variants) in Maple and experiments show that the latter two algorithms are significantly faster than the first. [ABSTRACT FROM AUTHOR]

VLSI technology makes the integration of multiple electronic components into a single electronic chip possible but Moore has already predicted that further miniaturization of VLSI chips will cause excess heat dissipation. Because of this impediment faced by this technology, the scientists have turned their focus to nano-technological alternatives to keep up the continuous device improvement and continue with the miniaturization procedure without facing any hurdle from the aspect of heat dissipation and space and time complexity. Memristor can be such a nano-alternative, which has the potential of very rapid execution, miniaturization and very low power consumption compared to transistor-based technology. The main objective of this paper is to show an application of symmetric function in cryptography. In order to do so, first, three-input and five-input symmetric functions are implemented in memristor-based nanocrossbar circuits. Then a dynamic circuit of symmetric functions is implemented where the input voltages convert the circuit to three-input or five-input symmetric function circuit as and when required. The simulation results show the satisfactory behavior of our proposed design. [ABSTRACT FROM AUTHOR]

We argue that the Nekrasov's partition function with two equivariant parameters is reproduced by the method of topological vertex, if we introduce a refined version of the vertex. Our refined topological vertex is expressed in terms the specialization of the Macdonald functions which is related to the equivariant character of the Hilbert scheme of ℂ2. We provide diagrammatic rules for computing the partition function. [ABSTRACT FROM AUTHOR]