Non-monic Polynomials Research Articles

Identifiability of diffusively coupled linear networks with partial instrumentation*

This paper presents identifiability conditions for identifying the complete dynamics of diffusively coupled linear networks. These conditions are derived by exploiting the uniqueness of the nonmonic polynomial network description, given the locations of the actuators and sensors. The analysis is performed under a more relaxed instrumentation setup than the typical restriction to a full set of sensors (full measurement) or a full set of actuators (full excitation). This leads to more general identifiability conditions, including more flexible instrumentation requirements.

Robust Stability and Stable Member Problems for Multilinear Systems

In this paper, we consider robust stability and stable member problems for linear systems whose characteristic polynomials are nonmonic polynomials with multilinear uncertainty. For both problems, the results are given by using the reflection (box) coefficients and the extreme point property of multilinear functions defined on the box. Finding stable member in a polynomial family is one of the hard problems of linear control theory. This issue is considered by visualizing the cases n-l=2 and n-l=3. Necessary and sufficient conditions for robust stability and the existence of a stable member of the multilinear polynomial family using the reflection coefficients are obtained. Several examples are provided.

Factoring non-monic polynomials represented by black boxes

We aim to factor a sparse polynomial a ∈ Z[ x 1 , ···, x n ] represented by a black box. The authors have previously developed efficient sparse Hensel lifting algorithms for the monic and square-free case that outperforms the algorithm by Kaltofen and Trager in 1990. We complete this black box factorization problem for the non-monic case with a new algorithm that computes the factors of a using many non-monic bivariate Hensel lifts. Our algorithm handles all cases of input a ∈ Z[ x 1 , ···, x n ] including the non-square-free and the non-primitive cases. We have implemented the algorithm in Maple with all major subroutines coded in C for efficiency.

Quantitative Hilbert Irreducibility and Almost Prime Values of Polynomial Discriminants

Abstract We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert’s Irreducibility Theorem for degree $n$ polynomials $f$ with $\textrm {Gal}(f) \subseteq A_n$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $n$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $n$ number fields with almost prime discriminants.

Correlation of arithmetic functions over $$\mathbb {F}_q[T]$$

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on $\Delta$ in a manner which is consistent with the Hardy-Littlewood Conjecture. We obtain a saving of $q$ if we consider monic polynomials only and $\Delta$ is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in $\Delta$. This allows us to obtain additional saving from equidistribution results for $L$-functions. We do all this in a combinatorial framework that applies to more general arithmetic functions than the indicator function of irreducibles, including the M\"{o}bius function and divisor functions.

Irreducibility of Random Polynomials

ABSTRACTWe study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic polynomials. Our data support conjectures made by Odlyzko and Poonen and by Konyagin, and we formulate a universality heuristic and new conjectures that connect their work with Hilbert’s Irreducibility Theorem and work of van der Waerden. The data indicate that the probability that a random polynomial is reducible divided by the probability that there is a linear factor appears to approach a constant and, in the large-degree limit, this constant appears to approach 1. In cases where the model makes it impossible for the random polynomial to have a linear factor, the probability of reducibility appears to be close to the probability of having a nonlinear, low-degree factor. We also study characteristic polynomials of random matrices with + 1 and − 1 entries.

An equivalent of Kronecker's theorem for powers of an algebraic number and structure of linear recurrences of fixed length

After defining a notion of $\epsilon$-density, we provide for any real algebraic number $\alpha$ an estimate of the smallest $\epsilon$ such that for each $m>1$ the set of vectors of the form $(t,t\alpha,...,t\alpha^{m-1})$ for $t\in\R$ is $\epsilon$-dense modulo 1, in terms of the multiplicative Mahler measure $M(A(x))$ of the minimal integral polynomial $A(x)$ of $\alpha$, and independently of $m$. In particular, we show that if $\alpha$ has degree $d$ it is possible to take $\epsilon = 2^{[d/2]}/M(A(x))$. On the other hand using asymptotic estimates for Toeplitz determinants we show that for sufficiently large $m$ we cannot have $\epsilon$-density if $\epsilon$ is a fixed number strictly smaller than $1/M(A(x))$. As a byproduct of the proof we obtain a result of independent interest about the structure of the $\Z$-module of integral linear recurrences of fixed length determined by a non-monic polynomial.

On Octonionic Polynomials

We discuss the generalization of results on quaternionic polynomials to the octonionic polynomials. In contrast to the quaternions the octonionic multiplication is non-associative. This fact although introducing some difficulties nevertheless leads to some new results. For instance, the monic and non-monic polynomials do not have, in general, the same set of zeros.

Real and complex stability radii of polynomial matrices

Analytic expressions are derived for the complex and real stability radii of non-monic polynomial matrices with respect to an arbitrary stability region of the complex plane. Numerical issues for computing these radii for different perturbation structures are also considered with application to robust stability of Hurwitz and Schur polynomial matrices.

The index of nonmonic polynomials

We show that the index of nonmonic polynomials gives the same information on the ring of integers in a number field as monic polynomials. In particular, we generalize a result of M.-N. Gras by proving that almost all cyclic extensions of Q of prime degree cannot be generated by a (nonmonic) polynomial with index 1. More precisely, we prove that this index goes to infinity with the conductor.

Anti-Isospectral Transformations in Quantum Mechanics

In this letter we demonstrate that there exists a remarkable class of quantum models admitting Z2 anti-isospectral transformations which change the form of the potential and invert the sign of a certain finite set of energy levels. These transformations help one to construct interesting classes of non-monic polynomials which are mutually orthogonal on two different intervals with two different weight functions.

Properties of the entire set of Hurwitz polynomials and stability analysis of polynomial families

It is proved in this paper that all Hurwitz polynomials of order not less than n form two simply connected Borel cones in the polynomial parameter space. Based on this result, edge theorems for Hurwitz stability of general polyhedrons of polynomials and boundary theorems for Hurwitz stability of compact sets of polynomials are obtained. Both cases of families of polynomials with dependent and independent coefficients are considered. Different from the previous ones, our edge theorems and boundary theorems are applicable to both monic and nonmonic polynomial families and do not require the convexity or the connectivity of the set of polynomials. Moreover, our boundary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine. >

Two necessary conditions for a complex polynomial to be strictly Hurwitz and their applications in robust stability analysis

The necessary conditions for a complex polynomial to be strictly Hurwitz are reviewed and rigorously proved. Both necessary conditions have been extended to cover nonmonic polynomials instead of monic polynomials. Also, based on these two results, some necessary conditions for an interval polynomial to be stable in terms of being strictly Hurwitz are obtained. They can be used to quickly determine the instability of a complex interval polynomial family. Finally, their application to the study of robust stability, in the case where coefficient perturbation intervals are functions of a single parameter, is briefly discussed. >