Monomials Of Degree Research Articles

Faces of the signed Birkhoff polytopes

We study faces of the signed Birkhoff polytopes, denoted by $\Omega_n^{\pm}$. We describe its nonempty faces, $1$-dimensional faces, $2$-dimensional faces, and facets. Moreover, we study the diameter and Hamiltonian connectivity of the graph of $\Omega_n^{\pm}$. In the end, we show that the reduced Gröbner basis of the toric ideal of the signed Birkhoff polytope $\Omega_n^{\pm}$ with respect to the graded reverse lexicographic order induced by rank orders has square-free initial monomials of degree $\leq n$.

A general formula for the index of depth stability of edge ideals

By a classical result of Brodmann, the function d e p t h R / I t depthR/I^t is asymptotically a constant, i.e. there is a number s s such that d e p t h R / I t = d e p t h R / I s depthR/I^t = depthR/I^s for t > s t > s . One calls the smallest number s s with this property the index of depth stability of I I and denotes it by d s t a b ( I ) dstab(I) . This invariant remains mysterious till now. The main result of this paper gives an explicit formula for d s t a b ( I ) dstab(I) when I I is an arbitrary ideal generated by squarefree monomials of degree 2. That is the first general case where one can characterize d s t a b ( I ) dstab(I) explicitly. The formula expresses d s t a b ( I ) dstab(I) in terms of the associated graph. The proof involves new techniques which relate different topics such as simplicial complexes, systems of linear inequalities, graph parallelizations, and ear decompositions. It provides an effective method for the study of powers of edge ideals.

Probabilistic estimation of the algebraic degree of Boolean functions

The algebraic degree is an important parameter of Boolean functions used in cryptography. When a function in a large number of variables is not given explicitly in algebraic normal form, it is usually not feasible to compute its degree, so we need to estimate it. We propose a probabilistic test for deciding whether the algebraic degree of a Boolean function f is below a certain value k. If the degree is indeed below k, then f will always pass the test, otherwise f will fail each instance of the test with a probability dtk(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{dt}_k(f)$$\\end{document}, which is closely related to the average number of monomials of degree k of the polynomials which are affine equivalent to f. The test has a good accuracy only if this probability dtk(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{dt}_k(f)$$\\end{document} of failing the test is not too small. We initiate the study of dtk(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{dt}_k(f)$$\\end{document} by showing that in the particular case when the degree of f is actually equal to k, the probability will be in the interval (0.288788, 0.5], and therefore a small number of runs of the test will be sufficient to give, with very high probability, the correct answer. Exact values of dtk(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{dt}_k(f)$$\\end{document} for all the polynomials in 8 variables were computed using the representatives listed by Hou and by Langevin and Leander.

Monomial projections of Veronese varieties: New results and conjectures

In this paper, we consider the homogeneous coordinate rings A(Yn,d)≅K[Ωn,d] of monomial projections Yn,d of Veronese varieties parameterized by subsets Ωn,d of monomials of degree d in n+1 variables where: (1) Ωn,d contains all monomials supported in at most s variables and, (2) Ωn,d is a set of monomial invariants of a finite diagonal abelian group G⊂GL(n+1,K) of order d. Our goal is to study when K[Ωn,d] is a quadratic algebra and, if so, when K[Ωn,d] is Koszul or G-quadratic. For the family (1), we prove that K[Ωn,d] is quadratic when s≥⌈n+22⌉. For the family (2), we completely characterize when K[Ω2,d] is quadratic in terms of the group G⊂GL(3,K), and we prove that K[Ω2,d] is quadratic if and only if it is Koszul. We also provide large families of examples where K[Ωn,d] is G-quadratic.

Monomial Cremona transformations and toric polar maps

Given a birational map in the three dimensional projective space defined by monomials of degree d, we prove that its inverse is defined by monomials of degree at most .

Petrie symmetric functions

For any positive integer $k$ and nonnegative integer $m$, we consider the symmetric $G\left( k,m\right)$ defined as the sum of all monomials of degree $m$ that involve only exponents smaller than $k$. We call $G\left( k,m\right)$ a symmetric function in honor of Flinders Petrie, as the coefficients in its expansion in the Schur basis are determinants of Petrie matrices (and thus belong to $\left\{ 0,1,-1\right\} $ by a classical result of Gordon and Wilkinson). More generally, we prove a Pieri-like rule for expanding a product of the form $G\left( k,m\right) \cdot s_{\mu}$ in the Schur basis whenever $\mu$ is a partition; all coefficients in this expansion belong to $\left\{ 0,1,-1\right\} $. We also show that $G\left( k,1\right) ,G\left( k,2\right) ,G\left( k,3\right) ,\ldots$ form an algebraically independent generating set for the symmetric functions when $1-k$ is invertible in the base ring, and we prove a conjecture of Liu and Polo about the expansion of $G\left( k,2k-1\right)$ in the Schur basis.

Low dimensional flow polytopes and their toric ideals

The toric ideal of a d-dimensional flow polytope has an initial ideal generated by square-free monomials of degree at most d. The toric ideal of a flow polytope of dimension at most four has an initial ideal generated by square-free monomials of degree at most two, with the only exception of the four-dimensional Birkhoff polytope, whose toric ideal has an initial ideal generated by a square-free cubic monomial. The proof is based on a method to classify certain compressed flow polytopes, and a construction of a quadratic pulling triangulation of them. Along the way compressed flow polytopes are classified up to dimension four, and their Ehrhart polynomials are computed.

The truncated moment problem on the union of parallel lines

In this article we study the bivariate truncated moment problem (TMP) of degree 2k on the union of parallel lines. First we present an alternative proof of Fialkow's solution [28] to the TMP on the union of two parallel lines (TMP–2pl) using the solution of the truncated Hamburger moment problem (THMP). We add a new equivalent solvability condition, which is then used together with the THMP, to solve the TMP on the union of three parallel lines (TMP–3pl), our second main result of the article. Finally, we establish a sufficient condition for the existence of a solution to the TMP on the union of n parallel lines in the pure case, i.e. when the moment matrix Mk is of the highest possible rank, or equivalently the only column relations come from the union of n lines. The condition is based on the feasibility of a certain linear matrix inequality, corresponding to the extension of Mk by adding rows and columns indexed by some monomials of degree k+1. The proof is by induction on n, where n≥2 and for the base of induction n=2 we use the solution of the TMP–2pl.

Numerical integration on higher dimensional simplicial and curved finite elements

We present a formula which evaluates lower degree monomials over higher dimensional simplices by means of integration of higher degree monomials over an interval, triangle or tetrahedron. Further, we show how to apply some higher order quadrature formulae on curved elements using a one-to-one mapping from the reference simplicial element to a curved element.Finally, we demonstrate that the non-zero Jacobian does not imply that this mapping is one-to-one.

Almost-Reed–Muller Codes Achieve Constant Rates for Random Errors

This paper considers “ $\delta $ -almost Reed–Muller codes”, i.e., linear codes spanned by evaluations of all but a $\delta $ fraction of monomials of degree at most $d$ . It is shown that for any $\delta > 0$ and any $\varepsilon >0$ , there exists a family of $\delta $ -almost Reed–Muller codes of constant rate that correct $1/2- \varepsilon $ fraction of random errors with high probability. For exact Reed–Muller codes, the analogous result is not known and represents a weaker version of the longstanding conjecture that Reed–Muller codes achieve capacity for random errors (Abbe-Shpilka-Wigderson STOC ’15). Our proof is based on the recent polarization result for Reed–Muller codes, combined with a combinatorial approach to establishing inequalities between the Reed–Muller code entropies.

Gorenstein rings generated by strongly stable sets of quadratic monomials

We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two. We compute their Hilbert series in several cases, which also provides an answer to a question by Migliore and Nagel.

On convex-valued G-m-monomials with applications in stability theory

Using the Mazur–Orlicz theorem and some results about the Frechet functional equation, we consider functional equations related to generalized monomials of degree n. From these considerations, we give some results on existence of single-valuedness and selections for convex-valued maps satisfying functional inclusions. Also, the Diaz–Margolis fixed point alternative is applied to solve the stability problem for set-valued generalized monomials of degree n. Finally, several particular cases are discussed and some applications are given.

On the Betti numbers and Rees algebras of ideals with linear powers

An ideal I subset mathbb {k}[x_1, ldots , x_n] is said to have linear powers if I^k has a linear minimal free resolution, for all integers k>0. In this paper, we study the Betti numbers of I^k, for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for d=2, 3 or n-1, and the product of all ideals generated by s variables, for s=n-1 or n-2. We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

Restricted classes of veronese type ideals and algebras

We study ideals which are generated by monomials of degree [Formula: see text] in the polynomial ring in [Formula: see text] variables and which satisfy certain numerical side conditions regarding their exponents. Typical examples of such ideals are the ideals of Veronese type, squarefree Veronese ideals or [Formula: see text]-spread Veronese ideals. In this paper we focus on [Formula: see text]-bounded [Formula: see text]-spread Veronese ideals and on Veronese ideals of bounded support. Their powers as well as their fiber cone are also considered.

Gorenstein $T$-spread Veronese algebras

Let $S=K[x_1, x_2, \dots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. We fix integers $d$ and $t$. A monomial $x_{i_1}x_{i_2}\cdots x_{i_d}$ with $i_1\leq i_2\leq\dots \leq i_d$ is $t$-spread if $i_j-i_{j-1}\geq t$, for any $2\leq j\leq n$. Let $I_{n,d,t}$ be the ideal generated by all $t$-spread monomials of degree $d$ and let $K[I_{n,d,t}]$ be the toric algebra generated by the monomials $v$ with $v\in G(I_{n,d,t})$. This generalizes the classical (squarefree)Veronese algebras. The aim of this paper is to characterize the algebras $K[I_{n,d,t}]$ which are Gorenstein.

Enumerating Partial Latin Rectangles

This paper deals with different computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is given by a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leqslant 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leqslant s \leqslant n \leqslant 6$. Numerical results have been cross-checked where possible.

A New Cube Attack on MORUS by Using Division Property

MORUS is an authenticated encryption algorithm and one of the candidates in the CAESAR competition. Currently, the security of MORUS received extensive attention. In this paper, a new existence terms detection method in superpoly recovery phase in cube attack is proposed. More precisely, the upper bounding degree of superpoly is first estimated by using the cube attack based on the division property with Mixed Integer Linear Programming tool. Moreover, the $t$ t -degree monomials that may be involved in the superpoly are divided into two groups, where the elements of the first group can be directly determined without using the solver via the embedded property. Compared with previous methods, the time consumption by the solvers of our new method is reduced significantly. In particular, the truth table from only the existent terms can be used to recover the superpoly in the offline phase of the cube attack. Therefore, the time complexity of cube attack can be further reduced. As illustrative example, the security of the reduced-step variants of MORUS-640-128 against cube attack is evaluated by using this new method. It is demonstrated that the key recovery attacks can be applied to 6/7-step MORUS-640-128. Furthermore, some integral distinguishers of 7-step MORUS-640-128/MORUS-1280-256 are achieved.

Combinatorial algorithm for the computation of cyclically standard regular bracket monomials

The ring of covariants of binary form is isomorphic to the ring of regular symmetric bracket polynomials in homogenized roots, which are obtained by symmetrization of regular bracket polynomials. In this paper, we give an algorithm for computing all cyclically standard bracket monomials of arbitrary degree. These monomials form a basis of the vector space of the regular bracket polynomials of a given degree. We also characterize the elemental bracket monomial of degree 2 in terms of graph theory.

Minimal set of binomial generators for certain Veronese 3-fold projections

The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese 3-fold projections. More precisely, for any integer d≥4 and any d-th root e of 1 we denote by Xd the toric variety defined as the image of the morphism φTd:P3⟶Pμ(Td)−1 where Td are all monomials of degree d in k[x,y,z,t] invariant under the action of the diagonal matrix M(1,e,e2,e3). In this work, we describe a Z-basis of the lattice Lη associated to I(Xd) as well as a minimal binomial set of generators of the lattice ideal I(Xd)=I+(η).

On the Gauss algebra of toric algebras

Let A be a K-subalgebra of the polynomial ring $$S=K[x_1,\ldots ,x_d]$$ of dimension d, generated by finitely many monomials of degree r. Then, the Gauss algebra $${\mathbb {G}}(A)$$ of A is generated by monomials of degree $$(r-1)d$$ in S. We describe the generators and the structure of $${\mathbb {G}}(A)$$, when A is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree 2, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph G with one loop, the embedding dimension of $${\mathbb {G}}(A)$$ is bounded by the complexity of the graph G.