Random Polynomials Research Articles

Expected Energy of Zeros of Elliptic Polynomials

AbstractIn 2011, Armentano, Beltrán and Shub obtained a closed expression for the expected logarithmic energy of the random point process on the sphere given by the roots of random elliptic polynomials. We consider a different approach which allows us to extend the study to the Riesz energies and to compute the expected separation distance.

Bergman kernel functions associated to measures supported on totally real submanifolds

Abstract We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold 𝐾 in C n \mathbb{C}^{n} is of polynomial growth. For example, this holds in dimension one if 𝐾 is a finite union of transverse Jordan arcs in ℂ. Our bounds are sharp when 𝐾 is smooth. We give an application to the equidistribution of the zeros of random polynomials, which extends a result of Shiffman–Zelditch to the higher-dimensional setting.

The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation

Abstract We extend the free convolution of Brown measures of $R$-diagonal elements introduced by Kösters and Tikhomirov [ 28] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.

Matrix Continued Fractions Associated with Lattice Paths, Resolvents of Difference Operators, and Random Polynomials

Matrix Continued Fractions Associated with Lattice Paths, Resolvents of Difference Operators, and Random Polynomials

Sums of random polynomials with differing degrees

Let μ \mu and ν \nu be probability measures in the complex plane, and let p p and q q be independent random polynomials of degree n n , whose roots are chosen independently from μ \mu and ν \nu , respectively. Under assumptions on the measures μ \mu and ν \nu , the limiting distribution for the zeros of the sum p + q p+q was computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021), p. 124719] as n → ∞ n \to \infty . In this paper, we generalize and extend this result to the case where p p and q q have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of μ \mu and ν \nu , scaled by the limiting ratio of the degrees of p p and q q . Additionally, our approach provides a complete description of the limiting distribution for the zeros of p + q p + q for any pair of measures μ \mu and ν \nu , with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.

A machine learning approach for optimizing and accurate prediction of performance parameters for stacked nanosheet transistor

In this article, the possibilities of accurate prediction of wide range of parameters and optimizing the same through machine learning (ML) approach have been demonstrated for the multi stacked nanosheet transistor (NSFET). The machine is trained by the generated data of the tedious calibrated technology computer aided (TCAD) simulations. An innovative strategy is employed that combines ML with device simulations. Numerous devices are simulated with different geometric parameters like height, width, length and equivalent oxide thickness of the channel. The input, output, and CV characteristics are extrapolated from the simulation which is predicted by ML models. The DC, Analog and RF parameters are derived with a domain expertise approach. The device parameters are anticipated with actual values by the ML approach. Random forest regression, linear regression, polynomial regression and decision tree regression are employed for the prediction of the performance parameters. Random forest regression models provide significant R2 score with minimal error percentage. It indicates that TCAD-augmented ML can be considered an alternative to device simulation due to its reduction in computational cost. This work can also be treated as a benchmark for accurate prediction of the NSFET.

Galois groups of random additive polynomials

We study the distribution of the Galois group of a random q q -additive polynomial over a rational function field: For q q a power of a prime p p , let f = X q n + a n − 1 X q n − 1 + … + a 1 X q + a 0 X f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots +a_1X^q+a_0X be a random polynomial chosen uniformly from the set of q q -additive polynomials of degree n n and height d d , that is, the coefficients are independent uniform polynomials of degree deg ⁡ a i ≤ d \deg a_i\leq d . The Galois group G f G_f is a random subgroup of GL n ⁡ ( q ) \operatorname {GL}_n(q) . Our main result shows that G f G_f is almost surely large as d , q d,q are fixed and n → ∞ n\to \infty . For example, we give necessary and sufficient conditions so that SL n ⁡ ( q ) ≤ G f \operatorname {SL}_n(q)\leq G_f asymptotically almost surely. Our proof uses the classification of maximal subgroups of GL n ⁡ ( q ) \operatorname {GL}_n(q) . We also consider the limits: q , n q,n fixed, d → ∞ d\to \infty and d , n d,n fixed, q → ∞ q\to \infty , which are more elementary.

Random Polynomial Neural Networks: Analysis and Design.

In this article, we propose the concept of random polynomial neural networks (RPNNs) realized based on the architecture of polynomial neural networks (PNNs) with random polynomial neurons (RPNs). RPNs exhibit generalized polynomial neurons (PNs) based on random forest (RF) architecture. In the design of RPNs, the target variables are no longer directly used in conventional decision trees, and the polynomial of these target variables is exploited here to determine the average prediction. Unlike the conventional performance index used in the selection of PNs, the correlation coefficient is adopted here to select the RPNs of each layer. When compared with the conventional PNs used in PNNs, the proposed RPNs exhibit the following advantages: first, RPNs are insensitive to outliers; second, RPNs can obtain the importance of each input variable after training; third, RPNs can alleviate the overfitting problem with the use of an RF structure. The overall nonlinearity of a complex system is captured by means of PNNs. Moreover, particle swarm optimization (PSO) is exploited to optimize the parameters when constructing RPNNs. The RPNNs take advantage of both RF and PNNs: it exhibits high accuracy based on ensemble learning used in the RF and is beneficial to describe high-order nonlinear relations between input and output variables stemming from PNNs. Experimental results based on a series of well-known modeling benchmarks illustrate that the proposed RPNNs outperform other state-of-the-art models reported in the literature.

Expansions and Characterizations of Sieved Random Walk Polynomials

We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations $P_0(x)=1$, $P_1(x)=x$, $x P_n(x)=(1-c_n)P_{n+1}(x)+c_n P_{n-1}(x),$ $n\in\mathbb{N}$ with $(c_n)_{n\in\mathbb{N}}\subseteq(0,1)$. For every $k\in\mathbb{N}$, the $k$-sieved polynomials $(P_n(x;k))_{n\in\mathbb{N}_0}$ arise from the recurrence coefficients $c(n;k):=c_{n/k}$ if $k|n$ and $c(n;k):=1/2$ otherwise. A main objective of this paper is to study expansions in the Chebyshev basis $\{T_n(x)\colon n\in\mathbb{N}_0\}$. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version $\mathrm{D}_k$ of the Askey-Wilson operator $\mathcal{D}_q$. It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative and obtained from $\mathcal{D}_q$ by letting $q$ approach a $k$-th root of unity. However, for $k\geq2$ the new operator $\mathrm{D}_k$ on $\mathbb{R}[x]$ has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for $k$-sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator $\mathrm{A}_k$.

Random polynomials in several complex variables

Random polynomials in several complex variables

Comparative Analysis of Classification Models for Predicting Cancer Stage in a Chilean Cancer Center

This study aimed to develop a predictive model for cancer stage using data from a Chilean cancer registry. Several factors, including cancer type, patient age, medical history, and time delay between diagnosis and treatment, were examined to determine their association with cancer stage. Multiple supervised multi-class classification methods were tested, and the best-performing models were identified. The results showed that the random forest, SVM polynomial, and composite models performed well across different stages, although distinguishing between Stages II and III was more challenging. The most important features for predicting cancer stage were found to be cancer type, TNM variables, and diagnostic extension. Variables related to treatment timing and sequence also showed some importance. It was emphasized that the results of predictive models should be interpreted carefully to avoid overprediction or underprediction. Clinical context and additional information should be considered to enhance the accuracy of predictions. The small dataset and limitations in data availability posed challenges in accurately predicting cancer stage for different cancer types. Implementing the predictive model can have various benefits, including informing treatment decisions, assessing disease severity, and optimizing resource allocation. Further research and expansion of the model's scope were recommended to improve its performance and impact. Overall, the study emphasized the potential of predictive models in cancer staging and highlighted the need for ongoing advancements in this field.

Almost sure behavior of the critical points of random polynomials

AbstractLet be a sequence of independent and identically distributed complex random variables with common distribution and let be the associated random polynomial in . Kabluchko established the conjecture stated by Pemantle and Rivin that the empirical measure associated with the critical points of converges weakly in probability to the base measure . In this note, we establish that the convergence, in fact, holds in the almost sure sense. Our result positively answers a question raised by Kabluchko and formalized as a conjecture in the recent paper (Michelen and Vu [arXiv:2212.11867]).

Efficient Multiparty Fully Homomorphic Encryption With Computation Fairness and Error Detection in Privacy Preserving Multisource Data Mining

In this article, we address the problem of data privacy in multisource data mining. To do it, we present a new multiparty fully homomorphic encryption (MP-FHE) scheme, in which all participants are completely fair to perform the same computation. At first, the proposed MP-FHE scheme is divided into five stages (i.e., calculation, configuration, recombination, resharing, and reconstruction stage) to achieve the unified computation form of addition and multiplication. Meanwhile, random bivariate polynomials and commutative encryption are used to achieve the degree reduction of polynomials and the continuity of computation. Moreover, we prove that the scheme meets result consistency and program termination under the fail-stop adversary model. Especially, three kinds of error detection criteria are presented to find errors in three different stages (i.e., recombination, resharing, and reconstruction stage), which provides the monitor basis for the fail-stop adversary model. In addition, the MP-FHE scheme is applied into privacy preserving k-means clustering algorithm. Finally, we evaluate the computation and communication performance of our scheme from both theoretical and experimental aspects, and the evaluation results show that the scheme is efficient enough for multisource data mining.

Quantum‐Accelerated Algorithms for Generating Random Primitive Polynomials Over Finite Fields

AbstractPrimitive polynomials over finite fields are crucial resources with broad applications across various domains in computer science, including classical pseudo‐random number generation, coding theory, and post‐quantum cryptography. Nevertheless, the pursuit of an efficient classical algorithm for generating random primitive polynomials over finite fields remains an ongoing challenge. In this work, it shows how this problem can be solved efficiently with the help of quantum computers. Moreover, the designs of specific quantum circuits to implement them are also presented. The research paves the way for the rapid and real‐time generation of random primitive polynomials in diverse quantum communication and computation applications.

Expected Number of Roots of Random Orthogonal Polynomials

Expected Number of Roots of Random Orthogonal Polynomials

Estimate of Number of Zeros of Random Polynomials

Estimate of Number of Zeros of Random Polynomials

An Interesting Class of Non-Kac Random Polynomials

An Interesting Class of Non-Kac Random Polynomials

A Probabilistic Version of Eneström–Kakeya Theorem for Certain Random Polynomials

This paper presents a comprehensive exploration of a probabilistic adaptation of the Eneström–Kakeya theorem, applied to random polynomials featuring various coefficient distributions. Unlike the deterministic rendition of the theorem, our study dispenses with the necessity of any specific coefficient order. Instead, we consider coefficients drawn from a spectrum of sets with diverse probability distributions, encompassing finite, countable, and uncountable sets. Furthermore, we provide a result concerning the probability of failure of Schur stability for a random polynomial with coefficients distributed independently and identically as standard normal variates. We also provide simulations to corroborate our results.

Maximum Likelihood Estimation of Hierarchical Linear Models from Incomplete Data: Random Coefficients, Statistical Interactions, and Measurement Error

We consider two-level models where a continuous response R and continuous covariates C are assumed missing at random. Inferences based on maximum likelihood or Bayes are routinely made by estimating their joint normal distribution from observed data and . However, if the model for R given C includes random coefficients, interactions, or polynomial terms, their joint distribution will be nonstandard. We propose a family of unique factorizations involving selected “provisionally known random effects” u such that is normally distributed and u is a low-dimensional normal random vector; we approximate via adaptive Gauss-Hermite quadrature. For polynomial models, the approximation is exact but, in any case, can be made as accurate as required given sufficient computation time. The model incorporates random effects as explanatory variables, reducing bias due to measurement error. By construction, our factorizations solve problems of compatibility among fully conditional distributions that have arisen in Bayesian imputation based on the Gibbs Sampler. We spell out general rules for selecting u, and show that our factorizations can support fully compatible Bayesian methods of imputation using the Gibbs Sampler. Supplementary materials for this article are available online.

Random polynomial graphs for random Turán problems

AbstractBukh and Conlon used random polynomial graphs to give effective lower bounds on , where is the th power of a balanced rooted tree . We extend their result to give effective lower bounds on , which is the maximum number of edges in a ‐free subgraph of the random graph . Analogous bounds for generalized Turán numbers in random graphs are also proven.