language:
Find link is a tool written by Edward Betts.Longer titles found: Twisted polynomial ring (view)
searching for Polynomial ring 103 found (362 total)
alternate case: polynomial ring
Noncommutative projective geometry
(227 words)
[view diff]
exact match in snippet
view article
find links to article
{\displaystyle k\langle x,y\rangle /(yx-qxy)} More generally, the quantum polynomial ring is the quotient ring: k ⟨ x 1 , … , x n ⟩ / ( x i x j − q i j x j xShort integer solution problem (2,913 words) [view diff] exact match in snippet view article find links to article
the quotient polynomial ring R=Z[x]/(xn−1){\displaystyle R=\mathbb {Z} [x]/(x^{n}-1)} are cyclic: consider the quotient polynomial ring R=Z[x]/(xn−1){\displaystyleSmooth scheme (1,047 words) [view diff] exact match in snippet view article find links to article
defined by some equations g1 = 0, ..., gr = 0, where each gi is in the polynomial ring k[x1,..., xn]. The affine scheme X is smooth of dimension m over kChevalley–Shephard–Todd theorem (858 words) [view diff] exact match in snippet view article find links to article
invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections. In theNoetherian scheme (1,267 words) [view diff] exact match in snippet view article find links to article
(in fact zero-dimensional) is given by the following quotient of a polynomial ring with infinitely many generators. Q[x1,x2,x3,…](x1,x22,x33,…){\displaystyleRegular sequence (1,216 words) [view diff] exact match in snippet view article find links to article
elements. For example, x, y(1-x), z(1-x) is a regular sequence in the polynomial ring C[x, y, z], while y(1-x), z(1-x), x is not a regular sequence. ButBracket ring (429 words) [view diff] exact match in snippet view article find links to article
{n}{d}}} generates a polynomial ring K[Λ(n,d)] over a field K. There is a homomorphism Φ(n,d) from K[Λ(n,d)] to the polynomial ring K[xi,j] in nd indeterminatesMori domain (284 words) [view diff] exact match in snippet view article find links to article
and only if it is a Mori domain and completely integrally closed. A polynomial ring over a Mori domain need not be a Mori domain. Also, the complete integralElliptic algebra (79 words) [view diff] exact match in snippet view article find links to article
certain regular algebra of a Gelfand–Kirillov dimension three (quantum polynomial ring in three variables) that corresponds to a cubic divisor in the projectiveNagata's conjecture (194 words) [view diff] exact match in snippet view article find links to article
algebra, Nagata's conjecture states that Nagata's automorphism of the polynomial ring k[x,y,z] is wild. The conjecture was proposed by Nagata (1972) andIdempotent (ring theory) (2,175 words) [view diff] no match in snippet view article
In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element isCohomology of a stack (101 words) [view diff] exact match in snippet view article find links to article
theorem states that the cohomology ring of a classifying stack is a polynomial ring. l-adic sheaf smooth topology Gaitsgory, Dennis; Lurie, Jacob (2019)Zariski's finiteness theorem (118 words) [view diff] exact match in snippet view article find links to article
theorem gives a positive answer to Hilbert's 14th problem for the polynomial ring in two variables, as a special case. Precisely, it states: Given aExp algebra (497 words) [view diff] exact match in snippet view article find links to article
ring of Laurent polynomials in 1 variable, while the exp ring is a polynomial ring in countably many generators. For each element g of G introduce a countableHarmonic polynomial (914 words) [view diff] exact match in snippet view article find links to article
harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radial polynomials. Consider aIntegrally closed domain (1,924 words) [view diff] exact match in snippet view article find links to article
and any field). A unique factorization domain (in particular, any polynomial ring over a field, over the integers, or over any unique factorization domain)Complex cobordism (1,520 words) [view diff] exact match in snippet view article find links to article
the Lazard ring, cannot be torsion since L{\displaystyle L} is a polynomial ring. Thus, x{\displaystyle x} must be in the kernel.) John Milnor (1960)Scheme-theoretic intersection (759 words) [view diff] exact match in snippet view article find links to article
projective variety with the homogeneous coordinate ring S/I, where S is a polynomial ring. If H={f=0}⊂Pn{\displaystyle H=\{f=0\}\subset \mathbb {P} ^{n}} isLocally nilpotent derivation (4,130 words) [view diff] exact match in snippet view article find links to article
Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring. Over a field k{\displaystyle k} of characteristic zero, to give aDeviation of a poset (553 words) [view diff] exact match in snippet view article find links to article
algebraically closed field and consider the poset of ideals of the polynomial ring k[x] in one variable. Since the deviation of this poset is the KrullRepresentation on coordinate rings (642 words) [view diff] exact match in snippet view article find links to article
finite-dimensional representation of G) and the coordinate ring is a polynomial ring. The most important case is when X is a symmetric variety; i.e., theFree module (1,750 words) [view diff] exact match in snippet view article find links to article
}), a submodule of a free module is free. If R is commutative, the polynomial ring R[X]{\displaystyle R[X]} in indeterminate X is a free module with aNagata's conjecture on curves (348 words) [view diff] exact match in snippet view article find links to article
which asks whether the invariant ring of a linear group action on the polynomial ring k[x1, ..., xn] over some field k is finitely generated. Nagata publishedProj construction (3,371 words) [view diff] exact match in snippet view article find links to article
generated by finitely many elements of degree 1{\displaystyle 1} (e.g. a polynomial ring or a homogenous quotient of it), all quasicoherent sheaves on ProjS{\displaystyleCanonical basis (2,326 words) [view diff] exact match in snippet view article find links to article
refers to the standard basis defined by the Kronecker delta. In a polynomial ring, it refers to its standard basis given by the monomials, (Xi)i{\displaystyleTorus action (604 words) [view diff] exact match in snippet view article find links to article
Example: Let S=k[x0,…,xn]{\displaystyle S=k[x_{0},\dots ,x_{n}]} be a polynomial ring over an infinite field k. Let T=Gmr{\displaystyle T=\mathbb {G} _{m}^{r}}List of cohomology theories (1,734 words) [view diff] exact match in snippet view article find links to article
in dimension 2. For ko, the coefficient ring is the quotient of a polynomial ring on three generators, η in dimension 1, x4 in dimension 4, and v14 inResultant (7,445 words) [view diff] exact match in snippet view article find links to article
and B in a polynomial ring R[x],{\displaystyle R[x],} where R=k[y1,…,yn]{\displaystyle R=k[y_{1},\ldots ,y_{n}]} is itself a polynomial ring over a fieldOre algebra (226 words) [view diff] exact match in snippet view article find links to article
x_{s}]} be a commutative polynomial ring (with A=K{\displaystyle A=K} when s=0{\displaystyle s=0}). The iterated skew polynomial ring A[∂1;σ1,δ1]⋯[∂r;σr,δr]{\displaystyleZariski's lemma (1,212 words) [view diff] exact match in snippet view article find links to article
Indeed, by the normalization lemma, K is a finite module over the polynomial ring k[x1,…,xd]{\displaystyle k[x_{1},\ldots ,x_{d}]} where x1,…,xd{\displaystyleField of definition (1,617 words) [view diff] exact match in snippet view article find links to article
A k-algebraic set is the zero-locus in An(kalg) of a subset of the polynomial ring k[x1, ..., xn]. A k-variety is a k-algebraic set that is irreducibleIdeal theory (1,075 words) [view diff] exact match in snippet view article find links to article
finitely generated algebra over a field (that is, a quotient of a polynomial ring over a field) behave somehow nicer than those in a general commutativeRegular chain (1,335 words) [view diff] exact match in snippet view article find links to article
number of free variables in the regular chain. The variables in the polynomial ring R=k[x1,…,xn]{\displaystyle R=k[x_{1},\ldots ,x_{n}]} are always sortedPerfect ideal (243 words) [view diff] exact match in snippet view article find links to article
modern definition when I{\displaystyle I} is a homogeneous ideal in polynomial ring, but may differ otherwise. Macaulay used Hilbert functions to defineDifference algebra (926 words) [view diff] exact match in snippet view article find links to article
is a field is called a difference field extension. The difference polynomial ring K{y}=K{y1,…,yn}{\displaystyle K\{y\}=K\{y_{1},\ldots ,y_{n}\}} overMonoid ring (573 words) [view diff] exact match in snippet view article find links to article
of polynomials over R. The monoid Nn (with the addition) gives the polynomial ring with n variables: R[Nn] =: R[X1, ..., Xn]. If G is a semigroup, theAlmost holomorphic modular form (669 words) [view diff] exact match in snippet view article find links to article
1/Im(τ)). The ring of almost holomorphic modular forms of level 1 is a polynomial ring over the complex numbers in the three generators E 2 ( τ ) − 3 / πDifferential operator (3,409 words) [view diff] exact match in snippet view article find links to article
R⟨D,X⟩{\displaystyle R\langle D,X\rangle } be the non-commutative polynomial ring over R in the variables D and X, and I the two-sided ideal generatedRees decomposition (154 words) [view diff] exact match in snippet view article find links to article
introduced by David Rees (1956). Suppose that a ring R is a quotient of a polynomial ring k[x1,...] over a field by some homogeneous ideal. A Rees decompositionAlgebraic closure (963 words) [view diff] exact match in snippet view article find links to article
d=degree(fλ){\displaystyle d={\rm {degree}}(f_{\lambda })}. Let R be the polynomial ring over K generated by uλ,i{\displaystyle u_{\lambda ,i}} for all λ∈Λ{\displaystyleStanley decomposition (140 words) [view diff] exact match in snippet view article find links to article
by Richard Stanley (1982). Suppose that a ring R is a quotient of a polynomial ring k[x1,...] over a field by some ideal. A Stanley decomposition of RBuchberger's algorithm (770 words) [view diff] exact match in snippet view article find links to article
crude version of this algorithm to find a basis for an ideal I of a polynomial ring R proceeds as follows: Input A set of polynomials F that generatesPolynomial greatest common divisor (7,570 words) [view diff] exact match in snippet view article find links to article
computed in R, then they exist and may be computed in every multivariate polynomial ring over R. In particular, if R is either the ring of the integers or aGroup scheme (2,860 words) [view diff] exact match in snippet view article find links to article
of T. Over an affine base, one can construct it as a quotient of a polynomial ring in n2 + 1 variables by an ideal encoding the invertibility of the determinantQuaternary cubic (322 words) [view diff] exact match in snippet view article find links to article
32, 40, 100. The generators of degrees 8, 16, 24, 32, 40 generate a polynomial ring. The generator of degree 100 is a skew invariant, whose square is aFinite morphism (851 words) [view diff] exact match in snippet view article find links to article
the inclusion of A1 − 0 into A1 is not finite. (Indeed, the Laurent polynomial ring k[y, y−1] is not finitely generated as a module over k[y].) This restrictsBrown–Peterson cohomology (443 words) [view diff] exact match in snippet view article find links to article
BP∗(BP){\displaystyle {\text{BP}}_{*}({\text{BP}})} is isomorphic to the polynomial ring π∗(BP)[t1,t2,…]{\displaystyle \pi _{*}({\text{BP}})[t_{1},t_{2},\ldotsBasis theorem (85 words) [view diff] exact match in snippet view article find links to article
kinds. Hilbert's basis theorem, in algebraic geometry, says that a polynomial ring over a Noetherian ring is Noetherian. Low basis theorem, a particularKostant polynomial (1,930 words) [view diff] exact match in snippet view article find links to article
used by Chevalley to prove that the ring of invariants was itself a polynomial ring. A detailed account of Kostant polynomials was given by Bernstein,Hypersurface (1,322 words) [view diff] exact match in snippet view article find links to article
of n. This is the geometric interpretation of the fact that, in a polynomial ring over a field, the height of an ideal is 1 if and only if the idealHochschild homology (3,119 words) [view diff] exact match in snippet view article find links to article
_{k}A}. One simple example is to compute the Hochschild homology of a polynomial ring of Q{\displaystyle \mathbb {Q} } with n{\displaystyle n}-generatorsBateman–Horn conjecture (1,047 words) [view diff] exact match in snippet view article find links to article
32{\frac {x}{(\log x)^{2}}}.} When the integers are replaced by the polynomial ring F[u] for a finite field F, one can ask how often a finite set of polynomialsCohomology ring (561 words) [view diff] exact match in snippet view article find links to article
product of n copies of RP∞{\displaystyle \mathbb {R} P^{\infty }} is a polynomial ring in n variables with coefficients in F2{\displaystyle \mathbb {F} _{2}}GCD domain (1,012 words) [view diff] exact match in snippet view article find links to article
for a GCD domain R.[citation needed] If R is a GCD domain, then the polynomial ring R[X1,...,Xn] is also a GCD domain. R is a GCD domain if and only ifSemi-invariant of a quiver (1,360 words) [view diff] exact match in snippet view article find links to article
}^{-u}(g_{2}){\det }^{u}(B)} The ring of semi-invariants equals the polynomial ring generated by det, i.e. SI(Q,d)=k[det]{\displaystyle {\mathsf {SI}}(QJacobson ring (829 words) [view diff] exact match in snippet view article find links to article
of algebraic geometry is a special case of the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring. A generalInvariant of a binary form (2,346 words) [view diff] exact match in snippet view article find links to article
degree 4, 8, 12, 18. The generators of degrees 4, 8, 12 generate a polynomial ring, which contains the square of Hermite's skew invariant of degree 18Zeeman's comparison theorem (627 words) [view diff] exact match in snippet view article find links to article
theorem, which says the cohomology ring of a classifying space is a polynomial ring.[citation needed] First of all, with G as a Lie group and with Q{\displaystyleAssociated graded ring (895 words) [view diff] exact match in snippet view article find links to article
theorem implies that grU{\displaystyle \operatorname {gr} U} is a polynomial ring; in fact, it is the coordinate ring k[g∗]{\displaystyle k[{\mathfrakSchinzel's hypothesis H (1,687 words) [view diff] exact match in snippet view article find links to article
analogous conjecture with the integers replaced by the one-variable polynomial ring over a finite field is false. For example, Swan noted in 1962 (forFano variety (1,240 words) [view diff] exact match in snippet view article find links to article
Fano variety. This is the projective scheme associated to a graded polynomial ring whose generators have degrees a0,...,an. If this is well formed, inCarlitz exponential (361 words) [view diff] exact match in snippet view article find links to article
Carlitz module – an example of a Drinfeld module. We work over the polynomial ring Fq[T] of one variable over a finite field Fq with q elements. The completionTorsion-free abelian group (774 words) [view diff] exact match in snippet view article find links to article
generated countable example is given by the additive group of the polynomial ring Z[X]{\displaystyle \mathbb {Z} [X]} (the free abelian group of countableSchubert polynomial (1,390 words) [view diff] exact match in snippet view article find links to article
fixing all but a finite number of elements. They form a basis for the polynomial ring Z[x1,x2,…]{\displaystyle \mathbb {Z} [x_{1},x_{2},\ldots ]} in infinitelySchubert polynomial (1,390 words) [view diff] exact match in snippet view article find links to article
fixing all but a finite number of elements. They form a basis for the polynomial ring Z[x1,x2,…]{\displaystyle \mathbb {Z} [x_{1},x_{2},\ldots ]} in infinitelyAscending chain condition on principal ideals (888 words) [view diff] exact match in snippet view article find links to article
trivial.) An integral domain A satisfies (ACCP) if and only if the polynomial ring A[t] does. The analogous fact is false if A is not an integral domainFinitely generated algebra (1,044 words) [view diff] exact match in snippet view article find links to article
R{\displaystyle R}-algebra is of finite type, but the converse is false: the polynomial ring R[X]{\displaystyle R[X]} is of finite type but not finite. Finite algebrasCoherent sheaf (6,652 words) [view diff] exact match in snippet view article find links to article
Spec(k){\displaystyle \operatorname {Spec} (k)} associated to the polynomial ring k[x]{\displaystyle k[x]}, which is not coherent because k[x]{\displaystyleLexicographic order (3,275 words) [view diff] exact match in snippet view article find links to article
ISBN 3-540-57456-5. Zbl 0922.68073. Robbiano, L. (1985). Term orderings on the polynomial ring. In European Conference on Computer Algebra (pp. 513-517). SpringerMultiplicative function (2,436 words) [view diff] exact match in snippet view article find links to article
function originates from R. Vaidyanathaswamy (1931). Let A = Fq[X], the polynomial ring over the finite field with q elements. A is a principal ideal domainBerlekamp's algorithm (1,672 words) [view diff] exact match in snippet view article find links to article
_{p}}-subspace, and an explicit basis for it can be calculated in the polynomial ring Fp[x,y]/(f,g){\textstyle \mathbb {F} _{p}[x,y]/(f,g)} by computingAlternating polynomial (1,092 words) [view diff] exact match in snippet view article find links to article
subrepresentations of the action of the symmetric group on n letters on the polynomial ring in n variables. (Formally, the symmetric group acts on n letters, andIntegral closure of an ideal (651 words) [view diff] exact match in snippet view article find links to article
closed. Let R=k[X1,…,Xn]{\displaystyle R=k[X_{1},\ldots ,X_{n}]} be a polynomial ring over a field k. An ideal I in R is called monomial if it is generatedBehrend's trace formula (1,001 words) [view diff] exact match in snippet view article find links to article
space of a topological group), whose rational cohomology ring is a polynomial ring in one generator (Borel's theorem), but we shall not use this directlyProper morphism (2,768 words) [view diff] exact match in snippet view article find links to article
contrast, the ring of regular functions on the affine line over k is the polynomial ring k[x], which does not have finite dimension as a k-vector space. ThereWu's method of characteristic set (1,460 words) [view diff] exact match in snippet view article find links to article
it may not be its system of generators. Let R be the multivariate polynomial ring k[x1, ..., xn] over a field k. The variables are ordered linearly accordingCharacter variety (1,186 words) [view diff] exact match in snippet view article find links to article
\mathbb {C} ^{3}}; its coordinate ring is isomorphic to the complex polynomial ring in 3 variables, C[x,y,z]{\displaystyle \mathbb {C} [x,y,z]}. RestrictingJacobian conjecture (1,872 words) [view diff] exact match in snippet view article find links to article
algebraically closed field of characteristic 0. Let k[X] denote the polynomial ring k[X1, ..., Xn] and k[F] denote the k-subalgebra generated by f1, .Crystal base (1,865 words) [view diff] exact match in snippet view article find links to article
polynomials g(q){\displaystyle g(q)} and h(q){\displaystyle h(q)} in the polynomial ring Q[q]{\displaystyle \mathbb {Q} [q]} such that h(0)≠0{\displaystyleSiegel modular form (1,629 words) [view diff] exact match in snippet view article find links to article
are the same as level 1 modular forms. The ring of such forms is a polynomial ring C[E4,E6] in the (degree 1) Eisenstein series E4 and E6. For degreeLyndon word (2,745 words) [view diff] exact match in snippet view article find links to article
algebra, and generate it; thus, the shuffle algebra is isomorphic to a polynomial ring over k, with the indeterminates corresponding to the Lyndon words.Tropical geometry (3,388 words) [view diff] exact match in snippet view article find links to article
+a_{ns}X_{n}\}.\end{aligned}}} Given a polynomial f in the Laurent polynomial ring K[x1±1,…,xn±1]{\displaystyle K[x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1}]}Splitting of prime ideals in Galois extensions (2,422 words) [view diff] exact match in snippet view article find links to article
the (finite) residue field OK/P. Suppose that h(X) factorises in the polynomial ring F[X] as h(X)=h1(X)e1⋯hn(X)en,{\displaystyle h(X)=h_{1}(X)^{e_{1}}\cdotsContinuous functional calculus (4,133 words) [view diff] exact match in snippet view article find links to article
elements with a∗a=aa∗{\displaystyle a^{*}a=aa^{*}}, is necessary, as the polynomial ring C[z,z¯]{\displaystyle \mathbb {C} [z,{\overline {z}}]} is commutativeInversive congruential generator (2,027 words) [view diff] exact match in snippet view article find links to article
f(x)=x2−cx−a∈Fq[x]{\displaystyle f(x)=x^{2}-cx-a\in \mathbb {F} _{q}[x]} (polynomial ring over Fq{\displaystyle \mathbb {F} _{q}}) is primitive. This is notHilbert scheme (3,242 words) [view diff] exact match in snippet view article find links to article
corresponds to a graded ideal IX∙{\displaystyle I_{X}^{\bullet }} of the polynomial ring S{\displaystyle S} in n+1{\displaystyle n+1} variables, with gradedNakayama's lemma (3,463 words) [view diff] exact match in snippet view article find links to article
M=0{\displaystyle M=0}. Of particular importance is the case that R is a polynomial ring with the standard grading, and M is a finitely generated module. TheOrdinal arithmetic (4,725 words) [view diff] exact match in snippet view article find links to article
factorization into primes under the natural product. While the full polynomial ring does have unique factorization, the subset of polynomials with non-negativePuiseux series (5,289 words) [view diff] exact match in snippet view article find links to article
numbersis an algebraically closed field that contains the univariate polynomial ring with complex coefficients. The Newton–Puiseux theorem is not validLinear algebraic group (5,949 words) [view diff] exact match in snippet view article find links to article
corresponding to the multiplicative group Gm = GL(1) is the Laurent polynomial ring k[x, x−1], with comultiplication given by x↦x⊗x.{\displaystyle x\mapstoAzumaya algebra (3,007 words) [view diff] exact match in snippet view article find links to article
σ∈Gal(L/K){\displaystyle \sigma \in {\text{Gal}}(L/K)} there is a twisted polynomial ring L[x]σ{\displaystyle L[x]_{\sigma }}, also denoted A(σ,b){\displaystylePermutation polynomial (2,515 words) [view diff] exact match in snippet view article find links to article
isomorphic to the general linear group GL(r, Fq). If g(x) is in the polynomial ring Fq[x] and g(xs) has no nonzero root in GF(q) when s divides q − 1,Glossary of module theory (2,569 words) [view diff] exact match in snippet view article find links to article
Quillen–Suslin theorem states that a finite projective module over a polynomial ring is free. quotient Given a left R{\displaystyle R}-module M{\displaystyleProjective variety (7,219 words) [view diff] exact match in snippet view article find links to article
on a projective variety. By definition, any homogeneous ideal in a polynomial ring yields a projective scheme (required to be prime ideal to give a variety)Divisor (algebraic geometry) (6,462 words) [view diff] exact match in snippet view article
on X. Let k be a field, and let n be a positive integer. Since the polynomial ring k[x1, ..., xn] is a unique factorization domain, the divisor classKoszul complex (5,003 words) [view diff] exact match in snippet view article find links to article
Xd{\displaystyle X_{1},X_{2},\dots ,X_{d}} are indeterminates and R is the polynomial ring k[X1,X2,…,Xd]{\displaystyle k[X_{1},X_{2},\dots ,X_{d}]}, the KoszulManin matrix (4,017 words) [view diff] exact match in snippet view article find links to article
} Observation 1. Coaction on a plane. Consider the polynomial ring C[x1, x2], and assume that the matrix elements a, b, c, d commute withAlgebraic K-theory (10,279 words) [view diff] exact match in snippet view article find links to article
a four-term exact sequence relating K0 of a ring R to K1 of R, the polynomial ring R[t], and the localization R[t, t−1]. Bass recognized that this theoremCayley–Hamilton theorem (10,190 words) [view diff] exact match in snippet view article find links to article
Euclidean division can in fact be performed within that commutative polynomial ring, and of course it then gives the same quotient B and remainder 0 asBergman's diamond lemma (2,695 words) [view diff] exact match in snippet view article find links to article
A=k\langle x,y,z\rangle /(yx-pxy,zx-qxz,zy-ryz)}, which is the quantum polynomial ring in 3 variables, and assume x<y<z{\displaystyle x<y<z}. Take <{\displaystyleTopological data analysis (11,069 words) [view diff] exact match in snippet view article find links to article
parameters as a Zn{\displaystyle \mathbb {Z} ^{n}} graded module over a polynomial ring in n variables. Tools from commutative and homological algebra areJordan–Chevalley decomposition (5,675 words) [view diff] exact match in snippet view article find links to article
d_{i}=\dim V_{i}}. Now, the Chinese remainder theorem applied to the polynomial ring k[t]{\displaystyle k[t]} gives a polynomial p(t){\displaystyle p(t)}Timeline of manifolds (1,178 words) [view diff] exact match in snippet view article find links to article
Sergei Novikov The ring of cobordism classes of stably complex manifolds is a polynomial ring on infinitely many generators of positive even degrees.