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Let $k$ be a field of characteristic two. We prove that a non constant monic polynomial $f\in k[X]$ of degree $n$ is the minimal/characteristic polynomial of a symmetric matrix with entries in $k$ if and only if it is not the product of pairwise distinct inseparable irreducible polynomials. In this case, we prove that $f$ is the minimal polynomial of a symmetric matrix of size $n$. We also prove that any element $\alpha\in k_{alg}$ of degree $n\geq 1$ is the eigenvalue of a symmetrix matrix of size $n$ or $n+1$, the first case happening if and only if the minimal polynomial of $\alpha$ is not the product of pairwise distinct inseparable irreducible polynomials.

The conditions for the unique solvability of the boundary-value problem for a functional differential equation with shifted and compressed arguments are expressed via the spectral radius formula for the corresponding class of functional operators. The use of this formula involves calculation of certain type limits, which, even in the simplest cases, exhibit an amazing “chaotic” dependence on the compression ratio. For example, it turns out that the spectral radius of the operator $$L_{2}(\mathbb{R}^{n})\ni u(x)\mapsto u(p^{-1}x+h)-u(p^{-1}x-h)\in L_{2}(\mathbb{R}^{n}),\quad p>1,\quad h\in\mathbb{R}^{n},$$ is equal to $$2p^{n/2}$$ for transcendental values of $$p$$ , and depends on the coefficients of the minimal polynomial for $$p$$ in the case where $$p$$ is an algebraic number. In this paper, we study this dependence. The starting point is the well-known statement that, given a velocity vector with rationally independent coordinates, the corresponding linear flow is minimal on the torus, i.e., the trajectory of each point is everywhere dense on the torus. We prove a version of this statement that helps to control the behavior of trajectories also in the case of rationally dependent velocities. Upper and lower bounds for the spectral radius are obtained for various cases of the coefficients of the minimal polynomial for $$p$$ . The main result of the paper is the exact formula of the spectral radius for rational (and roots of any degree of rational) values of $$p$$ .

Let $\mathbb{F}_q(T)$ be the field of rational functions in one variable over a finite field. We introduce the notion of a totally $T$-adic function: one that is algebraic over $\mathbb{F}_q(T)$ and whose minimal polynomial splits completely over the completion $\mathbb{F}_q(\!(T)\!)$. We give two proofs that the height of a nonconstant totally $T$-adic function is bounded away from zero, each of which provides a sharp lower bound. We spend the majority of the paper providing explicit constructions of totally $T$-adic functions of small height (via arithmetic dynamics) and minimum height (via geometry and computer search). We also execute a large computer search that proves certain kinds of totally $T$-adic functions of minimum height over $\mathbb{F}_2(T)$ do not exist. The problem of whether there exist infinitely many totally $T$-adic functions of minimum positive height over $\mathbb{F}_q(T)$ remains open. Finally, we consider analogues of these notions under additional integrality hypotheses., 25 pages; source code for computations in the paper available at https://github.com/RationalPoint/T-adic

Polynomials and matrices have played a very important role in the development of different branches of mathematics. Indeed, several mathematicians have introduced classical polynomials very useful for the scientific community such as the Lagrange’s interpolation polynomials, the Chebyshev’s polynomials and the Bernstein’s polynomials [1,2]. Also, there is a strong link between polynomials and matrices through the notions of the determinant, the characteristic polynomial and the minimal polynomial. Similarly, we will introduce in this article two polynomials which we will call twin polynomials as well as a matrix called kernels matrix. Finally, we will present some applications such as the resolution of recurrent sequences with second member and the establishment of several sum formulas.

Introduction: When diagnosing the deviations of controllable dynamic system parameters, it is convenient in terms of control simplicity to apply the Schreiber method which uses a set of rectangular pulses of equal duration as a test signal. Since for a single object you can construct many test signals which differ in the number of pulses, the problem arises how to minimize the number of test pulses when using the Schreiber method. Purpose: Simplification of test control and diagnostics of linear controllable dynamic systems. Results: It has been shown that a set of test pulse amplitude vectors is a kernel of the controllability matrix of a discrete analogue of the object under test. The problem is formulated of finding the optimal length of a test pulse in order to minimize the number of pulses in the test signal. For a given pulse length, the pulse amplitudes of an optimal test signal are equal to the coefficients of the control vector minimal polynomial for the discrete analog of the object relative to its system matrix. The number of test pulses can be reduced by choosing the pulse duration calculated from the imaginary component of the object poles. In particular, if an object has at least one pair of complex-conjugate poles, the number of test pulses does not at least exceed the order of the object. An algorithm has been developed for calculating a test signal for linear controllable object FDI by the Schreiber method. The input to the algorithm is the system matrix of the object, and the output is the length of the test pulse and the pulse amplitude vector. The efficiency of the algorithm is illustrated by FDI for two technical objects. Practical relevance: The results of the study can be applied to static parameter FDI of controllable dynamical objects which allow a linear description in their state space.

A Gaussian Pisot number is an algebraic integer with modulus greater than one whose other conjugates, over the quadratic field Q ( − 1 ) , are of modulus less than one. Given a nonreal algebraic number α , with modulus greater than one, we prove that there is a nonzero complex number λ such that the series ∑ n ∈ N ( Re ( λ α n ) 2 + Im ( λ α n ) 2 ) converges (resp. such that the sequence ( { Re ( λ α n ) } , { Im ( λ α n ) } ) n ∈ N has a unique limit point), if and only if α is a Gaussian Pisot number (resp. α is a Gaussian Pisot number satisfying P ( 1 ) ≥ 2 or deg ( P ) = 2 , where P is the minimal polynomial of α over Q ).

In this paper we consider the problem of counting algebraic numbers $\alpha$ of fixed degree $n$ and bounded height $Q$ such that the derivative of the minimal polynomial $P_{\alpha}(x)$ of $\alpha$ is bounded, $|P_{\alpha}'(\alpha)| < Q^{1-v}$. This problem has many applications to the problems of the metric theory of Diophantine approximation. We prove that the number of $\alpha$ defined above on the interval $\left(-\frac12, \frac12\right)$ doesn't exceed $c_1(n)Q^{n+1-\frac{1}{7}v}$ for $Q>Q_0(n)$ and $1.4 \le v \le \frac{7}{16}(n+1)$. Our result is based on an improvement to the lemma on the order of zero approximation by irreducible divisors of integer polynomials from A. Gelfond's monograph "Transcendental and algebraic numbers". The improvement provides a stronger estimate for the absolute value of the divisor in real points which are located far enough from all algebraic numbers of bounded degree and height and it's based on the representation of the resultant of two polynomials as the determinant of Sylvester matrix for the shifted polynomials. Keywords: Diophantine approximation, Hausdorff dimension, transcendental numbers, resultant, Sylvester matrix, irreducible divisor, Gelfond's lemma.

This paper presents new structural sensitivity reanalysis formulations based on the polynomial-type extrapolation methods. In these formulations, the displacement vector of the modified structure is expressed in the form of the vector sequences based on the fixed-point iteration method. By using these vector sequences, the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) methods calculate the approximate displacement vector of the modified structure by solving reduced linear least-square problems. Based on the definitions of the MPE and RRE methods, two sensitivity reanalysis formulations are derived, in which the first- and second-order sensitivities of the modified structure are obtained by solving a set of the over-determined least-square problems with much smaller size than the complete set of equations of the exact sensitivity analyses. The performance of the proposed sensitivity reanalysis formulations is evaluated by using four structural sensitivity reanalysis problems under multiple modifications in their initial designs. The results obtained from the numerical test problems indicate that the proposed sensitivity reanalysis formulations approximate the first- and second-order sensitivities of the modified structure with a high level of accuracy and they are able to converge to the exact solutions.

An 1861 theorem of Hermite asserts that for every field extension E / F of degree 5 there exists an element of E whose minimal polynomial over F is of the form $$f(x) = x^5 + c_2 x^3 + c_4 x + c_5$$ for some $$c_2, c_4, c_5 \in F$$ . We give a new proof of this theorem using techniques of Galois cohomology, under a mild assumption on F.

The concept of directed strongly regular graphs (DSRG) was introduced by Duval in 1988 [3] . In the present paper, we use representation theory of finite groups in order to investigate the directed strongly regular Cayley graphs. We first show that a Cayley graph C ( G , S ) is not a directed strongly regular graph if S is a union of some conjugate classes of G. This generalizes an earlier result of Leif K. Jorgensen [7] on abelian groups. Secondly, by using induced representations, we have a look at the Cayley graph C ( N ⋊ θ H , N 1 × H 1 ) with N 1 ⊆ N and H 1 ⊆ H , determining its characteristic polynomial and its minimal polynomial. Based on this result, we generalize the semidirect product method of Art M. Duval and Dmitri Iourinski in [4] and obtain a larger family of directed strongly regular graphs. Finally, we construct some directed strongly regular Cayley graphs on dihedral groups, which partially generalize the earlier results of Mikhail Klin, Akihiro Munemasa, Mikhail Muzychuk, and Paul Hermann Zieschang in [8] . By using character theory, we also give the characterization of directed strongly regular Cayley graphs C ( D n , X ∪ X a ) with X ∩ X ( − 1 ) = ∅ .

Let $\alpha$ be an algebraic integer of degree $d$, which is reciprocal. The house of $\alpha$ is the largest modulus of its conjugates. We proved that $d$-th power of the house of reciprocal $\alpha$ has a limit point. We presented a property of antireciprocal hexanomials. We compute the minimum of the houses of all reciprocal algebraic integers of degree $d$ having the minimal polynomial which is a factor of a $D$-th degree reciprocal or antireciprocal polynomial with at most eight monomials, say $\mathrm{mr}(d)$, for $d$ at most 180, $D\le 1.5d$ and $D\le 210$. We show that it is not necessary to take into account unprimitive polynomials. The computations suggest several conjectures., Comment: 18 pages, 2 tabels, 1 figure

Given a zero-dimensional ideal I in a polynomial ring, many computations start by finding univariate polynomials in I. Searching for a univariate polynomial in I is a particular case of considering the minimal polynomial of an element in P/I. It is well known that minimal polynomials may be computed via elimination, therefore this is considered to be a "resolved problem". But being the key of so many computations, it is worth investigating its meaning, its optimization, its applications (e.g. testing if a zero-dimensional ideal is radical, primary or maximal). We present efficient algorithms for computing the minimal polynomial of an element of P/I. For the specific case where the coefficients are in Q, we show how to use modular methods to obtain a guaranteed result. We also present some applications of minimal polynomials, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality., Comment: This is a fully revised version. To be published in Journal of Symbolic Computation, special Issue on Symbolic Computation and Satisfiability Checking

The height of an algebraic number $\alpha$ is a measure of how arithmetically complicated $\alpha$ is. We say $\alpha$ is totally $p$-adic if the minimal polynomial of $\alpha$ splits completely over the field $\mathbb{Q}_p$ of $p$-adic numbers. In this paper, we investigate what can be said about the smallest nonzero height of a degree $3$ totally $p$-adic number. In particular, we provide an algorithm to determine, given a prime $p$, the smallest height of a degree $3$ totally $p$-adic number., Comment: 15 pages, to be published in the proceedings from Algorithmic Number Theory Symposium XIV

Axial algebras of Jordan type $\eta$ are commutative algebras generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta\not\in\{0,1\}$ is fixed, with restrictive multiplication rules. These properties generalize the Pierce decompositions for idempotents in Jordan algebras, where $\frac{1}{2}$ is replaced with $\eta$. In particular, Jordan algebras generated by idempotents are axial algebras of Jordan type $\frac{1}{2}$. If $\eta\neq\frac{1}{2}$ then it is known that axial algebras of Jordan type $\eta$ are factors of the so-called Matsuo algebras corresponding to 3-transposition groups. We call the generating idempotents {\it axes} and say that an axis is {\it primitive} if its adjoint operator has 1-dimensional 1-eigenspace. It is known that a subalgebra generated by two primitive axes has dimension at most three. The 3-generated case has been opened so far. We prove that any axial algebra of Jordan type generated by three primitive axes has dimension at most nine. If the dimension is nine and $\eta=\frac{1}{2}$ then we either show how to find a proper ideal in this algebra or prove that the algebra is isomorphic to certain Jordan matrix algebras., Comment: 24 pages

In the field of algebraic systems biology, the number of minimal polynomial models constructed using discretized data from an underlying system is related to the number of distinct reduced Grobner bases for the ideal of the data points. While the theory of Grobner bases is extensive, what is missing is a closed form for their number for a given ideal. This work contributes connections between the geometry of data points and the number of Grobner bases associated to small data sets. Furthermore we improve an existing upper bound for the number of Grobner bases specialized for data over a finite field.

For $\beta > 1$ a real algebraic integer ({\it the base}), the finite alphabets $\mathcal{A} \subset \mathbb{Z}$ which realize the identity $\mathbb{Q}(\beta) = {\rm Per}_{\mathcal{A}}(\beta)$, where ${\rm Per}_{\mathcal{A}}(\beta)$ is the set of complex numbers which are $(\beta, \mathcal{A})$-eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and maximal alphabets are defined. The maximal alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base $\beta$ and Lehmer's problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in $\mathbb{Q}(\beta)$, generalizing Schmidt's theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases $\gamma_s := \gamma_{n, m_1 , \ldots , m_s}$ such that $\gamma_{s}^{-1}$ is the unique root in $(0,1)$ of an almost Newman polynomial of the type $-1+x+x^n +x^{m_1}+\ldots+ x^{m_s}$, $n \geq 3$, $s \geq 1$, $m_1 - n \geq n-1$, $m_{q+1}-m_q \geq n-1$ for all $q \geq 1$. For $\beta > 1$ a reciprocal algebraic integer close to one, the poles of modulus $< 1$ of the dynamical zeta function of the $\beta$-shift $\zeta_{\beta}(z)$ are shown, under some assumptions, to be zeroes of the minimal polynomial of $\beta$., Comment: 17 pages, 7 figures, 29 references

In this paper, we propose a distributed state observer in which the data of measurements and state estimates are protected by homomorphic cryptosystem. The proposed observer network is composed of local observers, where each of them utilizes encrypted local measurement, encryption of plant input, and encrypted estimates transmitted from its neighbors. All the operations in the network are performed on encrypted data without decryption, and the full state is recovered in every local observer as an encrypted message. Assuming the characteristic polynomial and the minimal polynomial of the state matrix for the plant are the same, the parameters in the observers are chosen to be integers. This not only allows finitetime convergence for the state estimates, but also makes the encrypted dynamic observers operate for infinite time horizon.

In this paper, based on the preconditioners presented by Zhang [A new preconditioner for generalized saddle matrices with highly singular(1,1) blocks. Int J Comput Maths. 2014;91(9):2091-2101], we ...

For a prime p, let $${E_{P,{P^m}}} = \left\{ {\left( {\begin{array}{*{20}{c}} aamp;b \\ {{p^{m – 1}}c}amp;d \end{array}} \right)\left| {a,b,c \in {Z_p},d} \right. \in {Z_{{p^m}}}} \right\}$$ . We first establish a ring isomorphism from $${Z_{p,{p^m}}}$$ onto $${E_{p,{p^m}}}$$ . Then we provide a way to compute -d and d–1 by using arithmetic in Zp and $${Z_{{p^m}}}$$ , and characterize the invertible elements of $${E_{p,{p^m}}}$$ . Moreover, we introduce the minimal polynomial for each element in $${E_{p,{p^m}}}$$ and give its applications.

Wu recently proposed three types of irreducible polynomials for low-complexity bit-parallel multipliers over F 2 n . In this paper, we consider new classes of irreducible polynomials for low-complexity bit-parallel multipliers over F 2 n , namely, repeated polynomial (RP). The complexity of the proposed multipliers is lower than those based on irreducible pentanomials. A repeated polynomial can be classified by the complexity of bit-parallel multiplier based on RPs, namely, C1, C2 and C3. If we consider finite fields that have neither a ESP nor a trinomial as an irreducible polynomial when n ≤ 1000 , then, in Wu’s result, only 11 finite fields exist for three types of irreducible polynomials when n ≤ 1000 . However, in our result, there are 181, 232(52.4%), and 443(100%) finite fields of class C1, C2 and C3, respectively.

In this study, a new approach based on the polynomial-type extrapolation methods is applied to the approximate structural reanalysis under multiple and large changes in the initial design. In this approach, the sequences of approximate displacement of the modified structure are constructed by using a fixed-point iteration method. These sequences are then further analyzed by two polynomial-type vector extrapolation methods to find the approximate response of the modified structure more accurately, namely, the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE). Based on a single initial design, the MPE and RRE methods approximate the displacement vector of the modified structure by solving a least-squares problem which is much smaller than the original system of equations of the exact analysis. The accuracy and efficiency of this reanalysis approach is evaluated on three large scale structural reanalysis problems under multiple and large changes in their initial designs. The obtained reanalysis results demonstrate that the MPE and RRE methods not only yield accurate results, but also are computationally efficient.

The effectiveness of the projection minimal polynomial method for solving the problem of determining the number of sources of signals acting on an antenna array (AA) with an arbitrary configuration and their angular directions has been studied. The method proposes estimating the degree of the minimal polynomial of the correlation matrix (CM) of the input process in the AA on the basis of a statistically validated root-mean-square criterion. Special attention is paid to the case of the ultrashort sample of the input process when the number of samples is considerably smaller than the number of AA elements, which is important for multielement AAs. It is shown that the proposed method is more effective in this case than methods based on the AIC (Akaike’s Information Criterion) or minimum description length (MDL) criterion.

For generalized saddle point problems, we establish a new matrix splitting preconditioner and give the implementing process in detail. The new preconditioner is much easier to be implemented than the modified dimensional split (MDS) preconditioner. The convergence properties of the new splitting iteration method are analyzed. The eigenvalue distribution of the new preconditioned matrix is discussed and an upper bound for the degree of its minimal polynomial is derived. Finally, some numerical examples are carried out to verify the effectiveness and robustness of our preconditioner on generalized saddle point problems discretizing the incompressible Navier–Stokes equations.

Let F q be a field of q elements, where q is a power of an odd prime p. The polynomial f ( y ) ∈ F q [ y ] defined by f ( y ) : = ( 1 + y ) ( q + 1 ) / 2 + ( 1 − y ) ( q + 1 ) / 2 has the property that f ( 1 − y ) = ρ ( 2 ) f ( y ) , where ρ is the quadratic character on F q . This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity.

For the generalized saddle-point problems, firstly, we introduce a modified generalized relaxed splitting (MGRS) preconditioner to accelerate the convergence rate of the Krylov subspace methods. Based on a block-triangular splitting of the saddle-point matrix, secondly, we propose a modified block-triangular splitting (MBTS). This new preconditioner is easily implemented since it has simple block structure. The spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed, respectively. Moreover, we apply the MGRS and the MBTS preconditioners to three-dimensional linearized Navier–Stokes equations. Then we derive the quasi-optimal parameters of the MGRS and the MBTS preconditioners for two and three-dimensional Navier–Stokes equations, respectively. Finally, numerical experiments are illustrated to show the preconditioning effects of the two new preconditioners.

If $\alpha$ is a non-zero algebraic number, we let $m(\alpha)$ denote the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(\alpha)$. For fixed $\alpha\in \bar{\mathbb Q}$, the map $t\mapsto m_t(\alpha)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $\alpha$. It remains open to determine whether there is a sequence of elements $\alpha_n\in \bar{\mathbb Q}$ such that the number of exceptional points for $\alpha_n$ tends to $\infty$ as $n\to \infty$. We utilize a connection with the Fibonacci sequence to formulate a conjecture on the $t$-metric Mahler measures. If the conjecture is true, we prove that it is best possible and that it implies the the existence of rational numbers with as many exceptional points as we like. Finally, with some computational assistance, we resolve various special cases of the conjecture that constitute improvements to earlier results.

In this paper we completely characterize the existence of algebraic traveling wave solutions for the celebrated Kolmogorov–Petrovskii–Piskunov/Zeldovich equation. To do it, we find necessary and sufficient conditions in order that a polynomial linear differential equation has a polynomial solution and we classify all the Darboux polynomials of the planar system \(\dot{x} =y\), \(\dot{y} =-c/d y +f(x)(f'(x)+r)\) where f is a polynomial with \(\deg f \ge 2\), \(c,d>0\) and r are real constants. All results are of interest by themselves.

Given multiple univariate or multivariate real polynomials f 1 , … , f n and a desired real zero z, we investigate a problem of finding the nearest polynomial f ˜ to f 1 , … , f n having z as a zero. We use a pair of norms to define the nearest polynomial. We remark that the difficulty of the problem depends on the norm pairs and give computing methods when the norm pairs consist of the p-norm and the q-norm for ( p , q ) = ( 2 , 2 ) , ( 2 , ∞ ) , and ( ∞ , ∞ ) .

A well-known theorem by S. A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x;D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x;D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p > 0. The main idea of the proof introduces a new way of embedding differential polynomial rings into bigger rings, which we name platinum rings, plus a key part of the proof involves the solution of matrix theory-based problems.

We provide a list of new natural $\mathsf{VNP}$-intermediate polynomial families, based on basic (combinatorial) $\mathsf{NP}$-complete problems that are complete under parsimonious reductions. Over finite fields, these families are in $\mathsf{VNP}$, and under the plausible hypothesis $\mathsf{Mod}_p\mathsf{P} \not\subseteq \mathsf{P/poly}$, are neither $\mathsf{VNP}$-hard (even under oracle-circuit reductions) nor in $\mathsf{VP}$. Prior to this, only the Cut Enumerator polynomial was known to be $\mathsf{VNP}$-intermediate, as shown by B\"{u}rgisser in 2000. We next show that over rationals and reals, two of our intermediate polynomials, based on satisfiability and Hamiltonian cycle, are not monotone affine polynomial-size projections of the permanent. This augments recent results along this line due to Grochow. Finally, we describe a (somewhat natural) polynomial defined independent of a computation model, and show that it is $\mathsf{VP}$-complete under polynomial-size projections. This complements a recent result of Durand et al. (2014) which established $\mathsf{VP}$-completeness of a related polynomial but under constant-depth oracle circuit reductions. Both polynomials are based on graph homomorphisms. A simple restriction yields a family similarly complete for $\mathsf{VBP}$., Comment: A preliminary version to appear in CSR 2016

A preconditioner is proposed for the large and sparse linear saddle point problems, which is based on a low-order three-by-three block saddle point form. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial for the preconditioned matrix are discussed. Numerical results show that the optimal convergence behavior can be achieved when the new preconditioner is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.

Parametric polynomial surface is a fundamental element in CAD systems. Since the most of the classic minimal surfaces are represented by non-parametric polynomial, it is interesting to study the minimal surfaces represented in parametric polynomial form. Recently, Ganchev presented the canonical principal parameters for minimal surfaces. The normal curvature of a minimal surface expressed in these parameters determines completely the surface up to a position in the space. Based on this result, in this paper, we study the bi-quintic isothermal minimal surfaces. According to the condition that any minimal isothermal surface is harmonic, we can acquire the relationship of some control points must satisfy. Follow up, we obtain two holomorphic functions f(z) and g(z) which give the Weierstrass representation of the minimal surface. Under the constrains that the minimal surface is bi-quintic, f(z) and g(z) can be divided into two cases. One case is that f(z) is a constant and g(z) is a quadratic polynomial, and another case is that the degree of f(z) and g(z) are 2 and 1 respectively. For these two cases, we transfer the isothermal parameter to canonical principal parameter, and then compute their normal curvatures and analyze the properties of the corresponding minimal surfaces. Moreover, we study some geometric properties of the bi-quintic harmonic surfaces based on the Bezier representation. Finally, some numerical examples are demonstrated to verify our results.

The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes (i.e., defined by a hereditary graph property), or equivalently to F -free graphs for a given graph family F (i.e., graphs which are F -free for all F ∈ F ). A tool to extend the results which show that for given hereditary graph classes the WIS problem can be solved in polynomial time is given by the following easy proposition: For any graph family F , if WIS can be solved for F -free graphs in polynomial time, then WIS can be solved for K 1 + F -free graphs (i.e., graphs which are K 1 + F -free for all F ∈ F ) in polynomial time. The main result of this paper is the following: A sufficient condition to extend the above proposition to K 2 + F -free graphs, and more generally to l K 2 + F -free graphs for any constant l (i.e., graphs which are l K 2 + F -free for all F ∈ F ), is that F -free graphs are m -plausible for a constant m , i.e., that for any F -free graph G the family of those maximal independent sets I of G such that every vertex of G not in I has more than m neighbors in I can be computed in polynomial time. In this context a section is devoted to show that (for instance) chordal graphs are m -plausible for a constant m . The proof of the main result is based on the idea/algorithm introduced by Farber to prove that every 2 K 2 -free graph has O ( n 2 ) maximal independent sets (Farber, 1989), which directly leads to a polynomial time algorithm to solve WIS for 2 K 2 -free graphs through a dynamic programming approach, and on some extensions of that idea/algorithm.

Denote the alternating and symmetric groups of degree $n$ by $A_n$ and $S_n$ respectively. Consider a permutation $\sigma\in S_n$ all of whose nontrivial cycles are of the same length. We find the minimal polynomials of $\sigma$ in the ordinary irreducible representations of $A_n$ and $S_n$., Comment: 21 pages

Given a zero-dimensional ideal I in a polynomial ring, many algorithms start by finding univariate polynomials in~I, or by computing a lex-Groebner basis of~I. These are related to considering the minimal polynomial of an element in P/I, which may be computed using Linear Algebra from a Groebner Basis (for any term-ordering). In this tutorial we'll see algorithms for computing minimal polynomials, applications of modular methods, and then some applications, namely algorithms for computing radicals and primary decompositions of zero-dimensional ideals, and also for testing radicality and maximality. We'll also address a kind of opposite problem: given a "geometrical description'', such as a finite set of points, find the ideal of polynomials which vanish at it. We start from the original Buchberger-Moeller algorithm, and we show some developments. All this will be done with a special eye on the practical implementations, and with demostrations in CoCoA.

The circuit of parallel Chien search is the largest in the decoder for long Bose-Chaudhuri-Hocquenghem (BCH) codes with strong correction capability. It can be decomposed into two functional blocks. One is the shift block which stores and transforms an error locator polynomial. The rest consists of substitution circuits which substitute finite field values into the polynomial to find its roots, which indicate error locations. For circuit-size reduction of the substitution, an error locator polynomial is divided by a product of minimal polynomials. If there are common roots between a dividend and divisor, the remainder inherits the common roots. Hence, error locations can be found by testing the remainder with the lower degree than that of the error locator polynomial. Our new architecture needs additional operations, such as the division. However, for 122-error-correcting BCH codes over GF(215), our new architecture can reduce the circuit size of the 16-parallel Chien search by 17% compared with an optimized conventional architecture.

Geometrically smooth spline surfaces, generalized to include n -sided facets or configurations of n ≠ 4 quads, can exhibit a curious lack of additional degrees of freedom for modeling or engineering analysis when refined. This paper establishes a minimal polynomial degree for smooth constructions of multi-sided surfaces that guarantees more flexibility in all directions under refinement. Degree bi-4 is both necessary and sufficient for flexibility-increasing G 1 -refinability within a bi-quadratic C 1 spline complex. Sufficiency is proven by two alternative flexibly G 1 -refinable constructions exhibiting good highlight line distributions.

Let [Formula: see text] be an algebraic number field with [Formula: see text] in the ring [Formula: see text] of algebraic integers of [Formula: see text] and [Formula: see text] be the minimal polynomial of [Formula: see text] over the field [Formula: see text] of rational numbers. In 1977, Uchida proved that [Formula: see text] if and only if [Formula: see text] does not belong to [Formula: see text] for any maximal ideal [Formula: see text] of the polynomial ring [Formula: see text] (see [Osaka J. Math. 14 (1977) 155–157]). In this paper, we apply the above result to obtain some necessary and sufficient conditions involving the coefficients of [Formula: see text] for [Formula: see text] to equal [Formula: see text] when [Formula: see text] is a trinomial of the type [Formula: see text]. In the particular case when [Formula: see text], it is deduced that [Formula: see text] is an integral basis of [Formula: see text] if and only if either (i) [Formula: see text] and [Formula: see text] or (ii) [Formula: see text] divides [Formula: see text] and [Formula: see text].

Let R be a ring with an automorphism α and a derivation δ. In this article we provide necessary and sufficient conditions for a skew polynomial ring R[x;α] and differential polynomial ring R[x;δ] to be 2-primal. We compute the Jacobson radical and the set of unit elements of a 2-primal skew polynomial ring R[x;α] and differential polynomial ring R[x;δ]. Also we establish the lower bounds on the stable range of a 2-primal skew polynomial ring R[x;α] and differential polynomial ring R[x;δ]. As an application we show that if R is 2-primal then the nth Weyl algebra over R is 2-primal and in this case J(An(R))=An(Nil∗(R)). As a consequence, we extend and unify several known results of [4], [8], [10], [18], [19], and [22].

For the generalized saddle-point problems, based on a new block-triangular splitting of the saddle-point matrix, we introduce a relaxed block-triangular splitting preconditioner to accelerate the convergence rate of the Krylov subspace methods. This new preconditioner is easily implemented since it has simple block structure. The spectral property of the preconditioned matrix is analysed. Moreover, the degree of the minimal polynomial of the preconditioned matrix is also discussed. Numerical experiments are reported to show the preconditioning effect of the new preconditioner.

Let AK denote the ring of algebraic integers of an algebraic number field K=Q(θ) where the algebraic integer θ has minimal polynomial F(x)=xn+axm+b over the field Q of rational numbers with n=mt+u, t∈N, 0≤u≤m−1. In this paper, we characterize those primes which divide the discriminant of F(x) but do not divide [AK:Z[θ]] when u=0 or u divides m; such primes p are important for explicitly determining the decomposition of pAK into a product of prime ideals of AK in view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a, b, m, n for AK to be equal to Z[θ].

Let $$K=\mathbb {Q}(\theta )$$ be an algebraic number field with $$\theta $$ in the ring $$A_K$$ of algebraic integers of K and F(x) be the minimal polynomial of $$\theta $$ over the field $$\mathbb {Q}$$ of rational numbers. For a rational prime p, let $$ F(x)\equiv \phi _1(x)^{e_1}\ldots \phi _r(x)^{e_r}(\mod p)$$ be its factorization into a product of powers of distinct irreducible polynomials modulo p with $$\phi _i(x)\in \mathbb Z[x]$$ monic. Let $$i_p(F)$$ denote the highest power of p dividing $$[A_K:\mathbb {Z}[\theta ]]$$ and $$i_{\phi _j}$$ denote the $$\phi _j$$ -index of F defined by $$i_{\phi _j}(F)= (\deg \phi _j)N_j$$ , where $$N_j$$ is the number of points with integral entries lying on or below the $$\phi _j$$ -Newton polygon of F away from the axes as well as from the vertical line passing through the last vertex of this polygon. The Theorem of Index of Ore states that $$i_p(F)\ge \sum \nolimits _{j=1}^{r}i_{\phi _j}(F)$$ and equality holds if F(x) satisfies a certain condition called p-regularity. In this paper, we extend the above theorem to irreducible polynomials with coefficients from valued fields of arbitrary rank and give a necessary and sufficient condition so that equality holds in the analogous inequality thereby generalizing similar results for discrete valued fields obtained in Montes and Nart (J Algebra 146:318–334, 1992) and Khanduja and Kumar (J Pure Appl Algebra 218:1206–1218, 2014). The introduction of the notion of $$\phi _j$$ -index of F in the general case involves some new results which are of independent interest as well.

The semi implicit method for pressure linked equations (SIMPLE) preconditioner is further generalized to a class of SIMPLE-like (SL) preconditioners for solving saddle point problems from the steady Navier-Stokes equations. The SL preconditioners can be also viewed as a generalization of the relaxed deteriorated PSS (RDPSS) preconditioner proposed by Cao et al. (2015). Convergence analysis of the corresponding SL iteration is presented and the optimal iteration parameter is obtained by minimizing the spectral radius of the SL iteration matrix. Moreover, Krylov subspace acceleration of the SL preconditioning is studied. The SL preconditioned saddle point matrix is analyzed. Results about eigenvalue and eigenvector distributions and the minimal polynomial are derived. Numerical experiments from "leaky" two dimensional lid-driven cavity problems are given, which show the advantages of the SL preconditioned GMRES over the DPSS and RDPSS preconditioned ones for solving saddle point problems.

Given a digraph G, we propose a new method to find the recurrence equation for the number of vertices n_k of the k-iterated line digraph L_k(G), for k>= 0, where L_0(G) = G. We obtain this result by using the minimal polynomial of a quotient digraph pi(G) of G. We show some examples of this method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic digraphs. In the first case, our method gives the enumeration of the ternary length-2 squarefree words of any length.

Based on the modified relaxed splitting (MRS) preconditioner proposed by Fan and Zhu (Appl Math Lett 55:18–26, 2016), a new relaxed splitting preconditioner is proposed for the generalized saddle point problems arising from the incompressible Navier–Stokes equations. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are analyzed. Numerical results show that the proposed preconditioner is superior to the MRS preconditioner when they are used as preconditioners to accelerate the convergence rate of the Krylov subspace methods, such as GMRES.

Based on a new matrix splitting of the original coefficient matrix, a modified relaxed splitting (MRS) preconditioner for generalized saddle point problems from the incompressible Navier–Stokes equations is considered. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied. The proposed preconditioner is closer to the original matrix than the generalized relaxed splitting (GRS) preconditioner in the sense of certain norm, which straightforwardly results in a MRS iteration method. Finally, numerical results are given to demonstrate the theoretical analysis. The results show that this novel preconditioner is competitive with and more effective than some of the best existing preconditioners.

The paper narrows down the possible automorphisms of extremal even unimodular lattices of dimension 72. With extensive computations in Magma [7] using the very sophisticated algorithm for computing class groups of algebraic number fields written by Steve Donnelly it is shown that the extremal even unimodular lattice Γ 72 from [17] is the unique extremal even unimodular lattice of dimension 72 that admits a large automorphism, where a d × d matrix is called large, if its minimal polynomial has an irreducible factor of degree > d / 2 .

Let α be a totally positive algebraic integer, and define its absolute trace to be Tr ( α ) deg ( α ) , the trace of α divided by the degree of α. Elementary considerations show that the absolute trace is always at least one, while it is plausible that for any ϵ > 0 , the absolute trace is at least 2 − ϵ with only finitely many exceptions. This is known as the Schur–Siegel–Smyth trace problem. Our aim in this paper is to show that the Schur–Siegel–Smyth trace problem can be considered as a special case of a more general problem.

The continuous analysis, such as smoothness and uniform convergence, for polynomials and polynomial-like functions using differential operators have been studied considerably, parallel to the study of discrete analysis for these functions, using difference operators. In this work, for the difference operator ∇h with size h ＞ 0, we verify that for an integer m ≥ 0 and a strictly decreasing sequence hn converging to zero, a continuous function f(x) satisfying ∇ m+1 hn f(khn) = 0, for every n ≥ 1 and k ∈ Z, turns to be a polynomial of degree ≤ m. The proof used the polynomial convergence, and additionally, we investigated several conditions on convergence to polynomials.

A new approach to polynomial higher-order approximation (smoothing) based on the basic elements method (BEM) is proposed. A BEM polynomial of degree n is defined by four basic elements specified on a three-point grid: x0 + α < x0 < x0 + β, αβ