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In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following: Let A be a Noetherian local ring of Krull
Monomial_conjecture
Kaplansky's conjectures Köthe conjecture: if a ring has no nil ideal other than {0}, then it has no nil one-sided ideal other than {0}. Monomial conjecture on
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
and Seshadri showed that Demazure's conjecture (for classical groups) follows from their work on standard monomial theory, and Littelmann extended this
Demazure_conjecture
Orthogonal symmetric polynomial family
the Macdonald polynomials become the sums over W orbits, which are the monomial symmetric functions when the root system has type A. If we put q = tα and
Macdonald_polynomials
Proposed lower bound on the Mahler measure for polynomials with integer coefficients
is an integral multiple of a product of cyclotomic polynomials or the monomial x {\displaystyle x} , in which case M ( P ( x ) ) = 1 {\displaystyle {\mathcal
Lehmer's_conjecture
Theorem about complexity measures of Boolean functions
1}n→{0,1} is at least the square root of its degree, thus settling a conjecture posed by Nisan and Szegedy in 1992. The proof is notably succinct, given
Sensitivity_theorem
factor is a monomial in the ψ i {\displaystyle \psi _{i}} , the first Chern classes of the n cotangent line bundles, as in Witten's conjecture. Let a 1
Lambda_g_conjecture
and monomial symmetric functions. The so-called q,t-Kostka polynomials are the coefficients of a resulting transition matrix. Macdonald conjectured that
N!_conjecture
Sumset of a field subject to a specific polynomial restriction
over a field F {\displaystyle F} . Suppose that the coefficient of the monomial x 1 k 1 ⋯ x n k n {\displaystyle x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}} in
Restricted_sumset
normal or Cohen–Macaulay. Standard monomial theory can be used to prove Demazure's conjecture. Standard monomial theory proves the Kempf vanishing theorem
Standard_monomial_theory
Type of Dirichlet series associated to number field extensions
using positive integer coefficients are called monomial groups, and Taketa (1930) proved that all monomial group are solvable groups. Moreover, this is
Artin_L-function
Russian-Swedish mathematician
B. Shapiro, "Trees, parking functions, syzygies, and deformations of monomial ideals", Transactions of the American Mathematical Society 356 (8), pp
Boris_Shapiro
Iranian mathematician (born 1942)
This topic later found applications in local cohomology, in the monomial conjecture, and other branches of commutative algebra. Zakeri was born in Urmia
Hossein_Zakeri
Differential algebra
nonzero monomial that has degree deg(g)+deg(h). To find such a monomial, pick the one in g with the highest degree. If there are multiple such monomials, pick
Weyl_algebra
American mathematician
received his PhD in 1981 from Princeton University with thesis Arithmetic of Monomial Relations between the Periods of Abelian Varieties under the supervision
Don_Blasius
American mathematician (born 1943)
the 2026 class of Fellows of the American Mathematical Society. Monomial conjecture Hochster, Melvin (1975). Topics in the homological theory of modules
Melvin_Hochster
Theorem in transcendental number theory
eγ(N)) by assumption. Since the product is symmetric, for any τ∈SN the monomials xτ(1)h1⋯xτ(N)hN and x1h1⋯xNhN have the same coefficient in the expansion
Lindemann–Weierstrass_theorem
Algebraic structure
exponent vector of the monomial. For a less cumbersome notation, the abbreviation Xα=X1α1⋯Xnαn is often used. The degree of a monomial Xα, frequently denoted
Polynomial_ring
Field of mathematics using techniques from combinatorics and commutative algebra
polytopes to simplicial spheres, the g-conjecture, which was resolved in 2018 by Karim Adiprasito. Square-free monomial ideal in a polynomial ring and Stanley–Reisner
Combinatorial commutative algebra
Combinatorial_commutative_algebra
Mathematical Concept
number of) gluing maps of monomials in ψ {\displaystyle \psi } and κ {\displaystyle \kappa } classes. These pushforwards of monomials (hereafter called basic
Tautological_ring
German mathematician
introduced the Littelmann path model and used it to solve several conjectures in standard monomial theory and other areas. Littelmann was an invited speaker at
Peter_Littelmann
Mathematical ring
ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite
Stanley–Reisner_ring
monomials. Factor: An expression being multiplied. Linear factor: A factor of degree one. Coefficient: An expression multiplying one of the monomials
List_of_polynomial_topics
check this for monomials in the ei's. Now, a monomial of even degree commutes with every element. Therefore if either x or y is a monomial of even degree
Polynomial_identity_ring
Special type of Boolean function
of bent functions, such as the homogeneous ones or those arising from a monomial over a finite field, but so far the bent functions have defied all attempts
Bent_function
for known or naive algorithms for the two problems, and often they are monomials such as n 2 {\displaystyle n^{2}} . Then A {\displaystyle A} is said to
Fine-grained_reduction
Symmetric function invariant of graphs
{\displaystyle \lambda } a partition, let m λ {\displaystyle m_{\lambda }} be the monomial symmetric polynomial associated to λ {\displaystyle \lambda } . Consider
Chromatic_symmetric_function
formula, including the Witten conjecture, the Virasoro constraints, and the λ g {\displaystyle \lambda _{g}} -conjecture. It is generalized by the Gopakumar–Mariño–Vafa
ELSV_formula
K-stability for Fano manifolds was made by Gang Tian in 1997, in response to a conjecture of Shing-Tung Yau from 1993 that there should exist a stability condition
K-stability_of_Fano_varieties
Basic result of approximation theory
special case, states that a necessary and sufficient condition for the monomials x n , n ∈ S ⊂ N {\displaystyle x^{n},\quad n\in S\subset \mathbb {N} }
Müntz–Szász_theorem
Measure of polynomial height
{\displaystyle K_{n}} be the set of polynomials that are products of monomials ± z 1 c 1 … z n c n {\displaystyle \pm z_{1}^{c_{1}}\dots z_{n}^{c_{n}}}
Mahler_measure
|K|<(q+n−3n−1) Consider the set of monomials of the form x1d1x2d2…xndn of degree exactly q−2. There are exactly (q+n−3n−1) such monomials. Thus, there exists a nonzero
Polynomial method in combinatorics
Polynomial_method_in_combinatorics
Algorithm checking for prime numbers
a given number is prime or composite without relying on mathematical conjectures such as the generalized Riemann hypothesis. The proof is also notable
AKS_primality_test
kind is commutative. The Dickson polynomials with parameter α = 0 give monomials. Dn(x,0)=xn. The Dickson polynomials with parameter α = 1 are related
Dickson_polynomial
polynomial invariant under permutation and inversion of variables, with leading monomial xλ, and orthogonal with respect to the density ∏ 1 ≤ i < j ≤ n ( x i x
Koornwinder_polynomials
operations that are at the core of algebra today. He was first to define the monomials x {\displaystyle x} , x 2 {\displaystyle x^{2}} , x 3 {\displaystyle x^{3}}
Timeline_of_mathematics
Discrete analog of a derivative
finite-difference analogs involving f( x T−1 h ). For instance, the umbral analog of a monomial xn is a generalization of the above falling factorial (Pochhammer k-symbol)
Finite_difference
of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already
H-vector
Number of partitions of an integer
distributive law to the product. This expands the product into a sum of monomials of the form xa1x2a2x3a3⋯ for some sequence of coefficients ai, only finitely
Partition function (number theory)
Partition_function_(number_theory)
Statement about cubic curves in the projective plane
of points required to determine a curve of degree d is the number of monomials of degree d, minus 1 from projectivization. For the first few d these
Cayley–Bacharach_theorem
Mathematical formula for the number of Young tableaux
fλ=dimVλ=χλ(e)=⟨sλ,p1(n)⟩ The expansion of Schur functions in terms of monomial symmetric functions uses the Kostka numbers: sλ=∑μKλμmμ, Then the inner
Hook_length_formula
American mathematician (born 1937)
toroidal embedding theory, the geometric approach to varieties defined by monomials. With Dave Bayer he published a paper "What can be computed in algebraic
David_Mumford
V W X Y Z See also Abstract simplicial complex Addition chain Scholz conjecture Algebraic combinatorics Alternating sign matrix Almost disjoint sets Antichain
Index of combinatorics articles
Index_of_combinatorics_articles
Ukrainian mathematician
doi:10.1070/IM1967v001n06ABEH000625. Yang, Tse-Chung; Yu, Chia-Fu (2013). "Monomial, Gorenstein and Bass Orders". arXiv:1308.6017 [math.RA]. Drozd, Yu. A.;
Andrei_Roiter
Knot invariant
the knot. Since this is only unique up to multiplication by the Laurent monomial ± t n {\displaystyle \pm t^{n}} , one often fixes a particular unique form
Alexander_polynomial
Mathematical field
of the study of analytic singularity and proof by Ecalle of the Dulac conjectures. It constitutes a formal object, extending the field of exp-log functions
Transseries
Circle-like pointset in a geometric plane
not occur in larger planes). Specifically, there are three classes of (monomial type) hyperovals, the hyperconics (f(t) = t2), proper translation hyperovals
Oval_(projective_plane)
Branch of mathematics
extension of the basis field) if and only if the Gröbner basis for any monomial ordering is reduced to {1}. By means of the Hilbert series, one may compute
Algebraic_geometry
Numbers parameterizing ways to partition a set
Additionally, this formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0: Δkxn=∑j=0k(−1)k−j(kj)(x+j)n. Because the Bernoulli
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Branch of mathematics
the monomial, demonstrating its use in calculus. We could also interpret this equation as the first two terms of the Taylor expansion of the monomial. Infinitesimals
Deformation_(mathematics)
Algebraic study of differential equations
∀p,q∈ΘY, ∀θμ∈Θ:p≥q⇒θμp≥θμq. Each derivative has an integer tuple, and a monomial order ranks the derivative by ranking the derivative's integer tuple. The
Differential_algebra
American mathematician
Amitsur, S. A. (1965). "Generalized Polynomial Identities and Pivotal Monomials". Transactions of the American Mathematical Society. 114 (1): 210–226
Wallace_Smith_Martindale
{B} _{2}(K)/2c} where GM(K) is the subgroup of GL(K), consisting of monomial matrices, and BGM(K)+ is the Quillen's plus-construction. Moreover, let
Bloch_group
Algebra used in 2D conformal field theories and string theory
where n is negative), then we may write the operator product of such a monomial as a normally ordered product of divided power derivatives of fields (here
Vertex_operator_algebra
Set of matrices
coefficients. For instance, Littlewood polynomials have coefficients ±1 in the monomial basis. Researchers such as Kurt Mahler, Andrew Odlyzko, Bjorn Poonen and
Bohemian_matrices
Spectral sequence
denoted ι. The kernel K0 consists of all elements SqIι for admissible monomials SqI generating 𝒜2, hence we have a map ⨁I admissible𝒜2⋅SqIι→K0 and we
Adams_spectral_sequence
Algebra in algebraic topology
− 1 {\displaystyle 2p^{k}-1} ( k ≥ 0 ) {\displaystyle (k\geq 0)} . The monomial basis for A ∗ {\displaystyle A_{*}} then gives another choice of basis
Steenrod_algebra
Alternative mathematical ordering
symmetric functions, for example as in xy + yz + zx where writing the final monomial as xz would distract from the pattern. A substantial use of cyclic orders
Cyclic_order
Indian inventions
mathematician S.S. Shrikhande in 1959. Standard monomial theory, C. S. Seshadri introduced a concept named Standard Monomials in 1978. Bipyrazole Organic Crystals
List of Indian inventions and discoveries
List_of_Indian_inventions_and_discoveries
Theoretical attack on block ciphers
which increases the number of equations by multiplying them with all monomials of a certain degree. Complexity estimates showed that the XL attack would
XSL_attack
American mathematician
diophantine equations including Euler's sum of powers conjecture and equations between monomials, abstract algebra, lattice theory and residuated lattices
Morgan_Ward
Number line and triangular tiling's symmetry mathematical structure
group Sn. Concretely, the elements of the group may be represented by monomial matrices (matrices having one nonzero entry in every row and column) whose
Affine_symmetric_group
Group of symmetries of an n-dimensional hypercube
Sn. The group G(r,n) may be concretely represented as the group of n×n monomial matrices whose nonzero entries are complex rth roots of unity. For r>2
Hyperoctahedral_group
Group in group theory and physics
Poincaré–Birkhoff–Witt theorem, it is thus the free vector space generated by the monomials zjp1k1p2k2⋯pnknq1ℓ1q2ℓ2⋯qnℓn , where the exponents are all non-negative
Heisenberg_group
Concept in algebraic geometry
{\displaystyle K} with basis given by (the Čech cohomology classes of) the inverse monomials [ x 1 − t 1 ⋯ x n − t n ] {\displaystyle \left[x_{1}^{-t_{1}}\cdots
Local_cohomology
Nonlinear differential operator used to study conformal mappings
also clear from the fact that it is in triangular form for the basis of monomials. A flat pseudogroup Γ is said to be "defined by differential equations"
Schwarzian_derivative
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