Monomials Of Degree Research Articles

Distribution of Elements of Cosets of Small Subgroups and Applications

We obtain a series of estimates on the number of small integers and small order Farey fractions which belong to a given coset of a subgroup of order t of the group of units of the residue ring modulo a prime p, in the case when t is small compared to p. We give two applications of these results: to the simultaneous distribution of two high degree monomials x k1 and x k2 modulo p and to a question of J. Holden and P. Moree on fixed points of the discrete logarithm.

On analytic equivalence of functions at infinity

In this paper we define the relation of analytic equivalence of functions at infinity. We prove that if the Łojasiewicz exponent at infinity of the gradient of a polynomial f ∈ R [ x 1 , … , x n ] is greater or equal to k − 1 , then there exists ε > 0 such that for every polynomial P ∈ R [ x 1 , … , x n ] of degree less or equal to k, whose coefficients of monomials of degree k are less or equal ε, the polynomials f and f + P are analytically equivalent at infinity.

On the Stanley depth of squarefree Veronese ideals

Let K be a field and S=K[x 1,?,x n ]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth?(M), and conjectured that depth?(M)?sdepth?(M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when M=I/J with J?I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in S. In particular, if I n,d is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1?d?n<5d+4, then sdepth?(I n,d )=?(n?d)/(d+1)?+d, and if d?1 and n?5d+4, then d+3?sdepth?(I n,d )??(n?d)/(d+1)?+d.

Application of the fields of objects formed on the basis of unit tensor objects to describe the tensile and compression failure stresses of pine wood in the principal orthotropy plane

The experimental investigation into tension and compression for softwood along grain direction shows the linear influence of wood density on the tensile and compression failure stress. For other directions determined by grain angles 25°, 45°, 90° for compression and 30°, 45° and 90° for tension the influence of density on failure stress was not shown. The investigation into the damaged structure of wood allows identification of the correlation between the tensile and compression failure stress and various failure mechanisms occurring for these grain angles. In the work the author proposed a new approach to description of failure stresses of the anisotropic materials. Usually this problem is solved on the basis of the formulated criteria for plane or complex stress states. Nonetheless the description of failure stresses may be formed by mutually compatible fragmentary descriptions. In order to describe the tensile and compression failure stress the objects formed on the basis of the unit objects of second and fourth rank tensors were used. The four models based on these objects or field of these objects in the form of the second degree monomials were used to determine the failure surface of normal stresses versus grain angle and density. The density was introduced to descriptions by dependence of model parameters on density using the least square estimation.

First Nonlinear Syzygies of Ideals Associated to Graphs

Consider an ideal I ⊂ K[x 1,…, x n ], with K an arbitrary field, generated by monomials of degree two. Assuming that I does not have a linear resolution, we determine the step s of the minimal graded free resolution of I where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree s + 3, and we compute the corresponding graded Betti number β s, s+3. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.

Polar syzygies in characteristic zero: The monomial case

Given a set of forms f = { f 1 , … , f m } ⊂ R = k [ x 1 , … , x n ] , where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D ( f ) , whose elements are called differential syzygies of f . There is a distinct submodule P ⊂ Z coming from the polynomial relations of f through its transposed Jacobian matrix, the elements of which are called polar syzygies of f . We say that f is polarizable if equality P = Z holds. This paper is concerned with the situation where f are monomials of degree 2, in which case one can naturally associate to them a graph G ( f ) with loops and translate the problem into a combinatorial one. The main result is a complete combinatorial characterization of polarizability in terms of special configurations in this graph. As a consequence, we show that polarizability implies normality of the subalgebra k [ f ] ⊂ R and that the converse holds provided the graph G ( f ) is free of certain degenerate configurations. One main combinatorial class of polarizability is the class of polymatroidal sets. We also prove that if the edge graph of G ( f ) has diameter at most 2 then f is polarizable. We establish a curious connection with birationality of rational maps defined by monomial quadrics.

Cases of Equality for Refinements of Bernstein’s Inequality

We identify extremal polynomials for a class of known refinements of Bernstein’s polynomial inequality. For example, the inequality \(\mid zp^\prime (z) - p(z) + p(0)\mid +\mid p(0)\mid \le (n-1) \max_{\mid u\mid \le 1} \mid p(u)\mid,\qquad \mid z\mid \le 1,\) holds for any polynomial p of degree n ≥ 2, and equality holds for some z, ¦z¦ = 1, if and only if p is a monomial of degree n or else p ≢ 0.

An explicit representation as quasi-sum of squares of a polynomial generated by the AG inequality

An explicit representation of the difference (x1+· · ·+xn)ⁿ−nⁿx1· · · xn for all natural n ≥ 2 is given as a sum of pij(xi − xj)2 over all 1 ≤ i < j n where pij = pij(x1, . . . , xn) are homogeneous polynomials of degree n − 2 whose coefficients at all possible monomials of degree n − 2 are positive.

On Ideals Whose Radical Is a Monomial Ideal

ABSTRACT We develop a new general method for constructing a polynomial ideal that has the same radical as a given monomial ideal, but has fewer generators. This provides upper bounds for the arithmetical rank of monomial ideals. An explicit formula is given for ideals generated by squarefree monomials of degree 2.

Revisiting two theorems of Curto and Fialkow on moment matrices

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence y ∈ R Z + n y\in \mathbb {R}^{\mathbb {Z}^n_+} whose moment matrix M ( y ) M(y) is positive semidefinite and has finite rank r r is the sequence of moments of an r r -atomic nonnegative measure μ \mu on R n \mathbb {R}^n . We give an alternative proof for this result, using algebraic tools (the Nullstellensatz) in place of the functional analytic tools used in the original proof of Curto and Fialkow. An easy observation is the existence of interpolation polynomials at the atoms of the measure μ \mu having degree at most t t if the principal submatrix M t ( y ) M_t(y) of M ( y ) M(y) (indexed by all monomials of degree ≤ t \le t ) has full rank r r . This observation enables us to shortcut the proof of the following result. Consider a basic closed semialgebraic set F = { x ∈ R n ∣ h 1 ( x ) ≥ 0 , … , h m ( x ) ≥ 0 } F=\{x\in \mathbb {R}^n\mid h_1(x)\ge 0, \ldots ,h_m(x)\ge 0\} , where h j ∈ R [ x 1 , … , x n ] h_j\in \mathbb {R}[x_1,\ldots ,x_n] and d := max j = 1 m ⁡ ⌈ deg ⁡ ( h j ) / 2 ⌉ d:=\operatorname {max}_{j=1}^m \lceil \operatorname {deg}(h_j)/2\rceil . If M t ( y ) M_t(y) is positive semidefinite and has a flat extension M t + d ( y ) M_{t+d}(y) such that all localizing matrices M t ( h j ∗ y ) M_{t}(h_j\ast y) are positive semidefinite, then y y has an atomic representing measure supported by F F . We also review an application of this result to the problem of minimizing a polynomial over the set F F .

A note on the Rees algebra of a bipartite graph

Let R = K [ x 1 , … , x n ] be a polynomial ring over a field K and let I be an ideal of R generated by a set x α 1 , … , x α q of square-free monomials of degree two such that the graph G defined by those monomials is bipartite. We study the Rees algebra R ( I ) of I, by studying both the Rees cone R + A ′ generated by the set A ′ = { e 1 , … , e n , ( α 1 , 1 ) , … , ( α q , 1 ) } and the matrix C whose columns are the vectors in A ′ . It is shown that C is totally unimodular. We determine the irreducible representation of the Rees cone in terms of the minimal vertex covers of G. Then we compute the a-invariant of R ( I ) .

Symbolic powers of edge ideals

Let I⊂R=k[ X]=k[X 1,…,X n] be an ideal in a polynomial ring over the field k. We define the essential symbolic module of I to be the R/ I-module F(I)=⊕ r⩾2(I (r)/Σ r(I)), where Σ r(I)=∑ ı=1 r−1I (ı)I (r−ı) and I ( m) stands for the mth symbolic power of I. We will mainly focus on the case where I is generated by square-free monomials of degree two. Among our main results are optimal bounds for the degrees of the minimal generators of F(I) , several criteria for a monomial to be such a generator and an upper bound for the generation type of the symbolic Rees algebra of I. As a byproduct we recapture the result of Simis, Vasconcelos, and Villarreal on when such an ideal is normally torsionfree.

I-stable ideals

We define the concept of I-stable ideals in the ring of commutative polynomials over a field, generalizing the so-called stable ideals, which arise as ideals of higher terms under general linear changes of variables. The interest in ideals of this type is motivated by the fact that certain problems concerning homogeneous ideals (for example, the problem of obtaining upper estimates for the graded Betti numbers) can be reduced to the study of stable ideals. I-stable ideals retain many interesting properties of stable ideals. In particular, the minimal resolutions of I-stable ideals constructed in this paper enable us to obtain an explicit formula for the graded Betti numbers, which turn out to be independent of the characteristic of the ground field. Factor rings by I-stable ideals generated by monomials of degree ≥2 are Golod rings. We also consider other analogues of stable ideals (strongly and weakly I-stable ideals) and give conditions sufficient for the factor ring by an I-stable ideal to be Cohen-Macaulay or Gorenstein.

ON THE NORMALITY OF MONOMIAL IDEALS OF MIXED PRODUCTS

Let R = K[x, y] be a polynomial ring in two disjoint sets of variables x, y over a field K. We study ideals of mixed products L = IkJr + IsJt such that k + r = s + t, where Ik (resp. Jr ) denotes the ideal of R generated by the square-free monomials of degree k (resp. r) in the x (resp. y ) variables. Our main result is a characterization of when a given ideal L of mixed products is normal. *Partially supported by CONACyT grant 27931E and SNI, México.

On the density of the set of generators of a polynomial algebra

Let K [ X ] = K [ x 1 , . . . , x n ] , n ≥ 2 , K[X] = K[x_1,...,x_n], ~n \ge 2, be the polynomial algebra over a field K K of characteristic 0 0 . We call a polynomial p ∈ K [ X ] ~p \in K[X] coordinate (or a generator) if K [ X ] = K [ p , p 2 , . . . , p n ] K[X] = K[p, p_2, ..., p_n] for some polynomials p 2 , . . . , p n ~p_2, ..., p_n . In this note, we give a simple proof of the following interesting fact: for any polynomial h ~h~ of the form ( x i + q ) , ~(x_i + q), where q q is a polynomial without constant and linear terms, and for any integer m ≥ 2 ~m \ge 2 , there is a coordinate polynomial p ~p~ such that the polynomial ( p − h ) (p-h) has no monomials of degree ≤ m \leq m . A similar result is valid for coordinate k k -tuples of polynomials, for any k &gt; n k &gt; n . This contrasts sharply with the situation in other algebraic systems. On the other hand, we establish (in the two-variable case) a result related to a different kind of density. Namely, we show that given a non-coordinate two-variable polynomial, any sufficiently small perturbation of its non-zero coefficients gives another non-coordinate polynomial.

Bipartite Graphs Whose Edge Algebras Are Complete Intersections

Let R be a monomial subalgebra of k[x1,…,xN] generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose vertices are x1,…,xN and whose edges are {(xi,xj)|xixj∈R}. Conversely, for any graph G with vertices {x1,…,xN} we define the edge algebra associated with G as the subalgebra of k[x1,…,xN] generated by the monomials {xixj|(xi,xj) is an edge of G}. We denote this monomial algebra by k[G]. This paper describes all bipartite graphs whose edge algebras are complete intersections.

Generating monomials in dimensions three and four

We introduce a technique to determine a lower bound on the number of monomials of degree d − 1 in n variables needed to generate all monomials of degree d. For n = 3, this gives a new proof of a result by Curtis and for n = 4, it improves the lower bound of Geramita et al. (1986). We also give a cover for the dimension four case that improves the upper bound given in the paper by Geramita and others.

A property deducible from the generic initial ideal

Let S d be the vector space of monomials of degree d in the variables x 1,…, x s . For a subspace V ⊆ S d which is in general coordinates, consider the subspace gin( V) ⊆ S d generated by initial monomials of polynomials in V for the revlex order. We address the question of what properties of V may be deduced from gin( V). This is an approach for understanding what algebraic or geometric properties of a homogeneous ideal I ⊆ k[ x 1,…, x s ] that may be deduced from its generic initial ideal gin( I).

On the equations of the edge cone of a graph¶and some applications

Let G be a graph such that none of its components is bipartite. We describe the facets of the cone generated by the columns of the incidence matrix of G. Let k[G] be the subring generated by the monomials of degree two defining the edges of G, where k is a field. Some estimates for the a-invariant of k[G] are shown when G is the cone of a normal connected non bipartite graph or G is the join of two normal connected non bipartite graphs.

Chromatic polynomials and order ideals of monomials

One expansion of the chromatic polynomial π( G, x) of a graph G relies on spanning trees of a graph. In fact, for a connected graph G of order n, one can express π(G,x)=(−1) n−1x ∑ i=1 n−1 t i(1−x) i ), where t i is the number of spanning trees with external activity 0 and internal activity i. Moreover, it is known (via commutative ring theory) that t i arises as the number of monomials of degree n − i − 1 in a set of monomials closed under division. We describe here how to explicitly carry out this construction algebraically. We also apply this viewpoint to prove a new bound for the roots of chromatic polynomials.