Strictly positive measure

Given a graded commutative algebra finitely generated over a field, the Hilbert function, Hilbert polynomial, and Hilbert series are three strongly related notions which measure the growth of the dimension of its homogeneous components.

These notions have been extended to filtered algebras and graded filtered modules over these algebras.

The typical situations where these notions are used are the following:

• The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree.
• The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree.
• The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial.

The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space.

Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations.

Definitions and main properties

Let us consider a finitely generated graded commutative algebra S over a field K, which is finitely generated by elements of positive degree. This means that

$S=\bigoplus _{i\geq 0}S_{i}\$

and that $S_{0}=K$.

The Hilbert function

$HF_{S}\;:\;n\mapsto \dim _{K}\,S_{n}$

maps the integer n onto the dimension of the K-vector space S_n. The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series

$HS_{S}(t)=\sum _{n=0}^{\infty }HF_{S}(n)\,t^{n}.$

If S is generated by h homogeneous elements of positive degrees $d_{1},\ldots ,d_{h}$, then the sum of the Hilbert series is a rational fraction

$HS_{S}(t)={\frac {Q(t)}{\prod _{i=1}^{h}(1-t^{d_{i}})}}\,,$

where Q is a polynomial with integer coefficients.

If S is generated by elements of degrees 1 then the sum of the Hilbert series may be rewritten as

$HS_{S}(t)={\frac {P(t)}{(1-t)^{\delta }}}\,,$

where P is a polynomial with positive integer coefficients.

In this case the series expansion of this rational fraction is

$HS_{S}(t)=P(t)\,\left(1+\delta \,t+\cdots +{\binom {n+\delta -1}{\delta -1}}\,t^{n}+\cdots \right)$

where the binomial coefficient ${\binom {n+\delta -1}{\delta -1}}$ is $\;{\frac {(n+\delta -1)(n+\delta -2)\cdots n}{(\delta -1)!}}\;$ for $n>-\delta$ and 0 otherwise.

This shows that there exists a unique polynomial $HP_{S}(n)$ with rational coefficients which is equal to $HF_{S}(n)$ for $n\geq \deg P-\delta +1$. This polynomial is the Hilbert polynomial. The least n₀ such that $HP_{S}(n)=HF_{S}(n)$ for n ≥ n₀ is called the Hilbert regularity. It may be lower than $\deg P-\delta +1$.

The Hilbert polynomial is a numerical polynomial, since the dimensions are integers, but the polynomial almost never has integer coefficients Template:Harv.

All these definitions may be extended to finitely generated graded modules over S, with the only difference that a factor t^m appears in the Hilbert series, where m is the minimal degree of the generators of the module, which may be negative.

The Hilbert function, the Hilbert series and the Hilbert polynomial of a filtered algebra are those of the associated graded algebra.

The Hilbert polynomial of a projective variety V in P^{n is defined as the Hilbert polynomial of the homogeneous coordinate ring of V.}

Graded algebra and polynomial rings

Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g₁, ..., g_n of degree 1, then the map which sends X_i onto g_i defines an homomorphism of graded rings from $R_{n}=K[X_{1},\ldots ,X_{n}]$ onto S. Its kernel is a homogeneous ideal I and this defines an isomorphism of graded algebra between $R_{n}/I$ and S.

Thus, the graded algebras generated by elements of degree 1 are exactly, up to an isomorphism, the quotients of polynomial rings by homogeneous ideals. Therefore, the remainder of this article will be restricted to the quotients of polynomial rings by ideals.

Properties of Hilbert series

Additivity

Hilbert series and Hilbert polynomial are additive relatively to exact sequences. More precisely, if

$0\;\rightarrow \;A\;\rightarrow \;B\;\rightarrow \;C\;\rightarrow \;0$

is an exact sequence of graded or filtered modules, then we have

$HS_{B}=HS_{A}+HS_{C}$

and

$HP_{B}=HP_{A}+HP_{C}.$

This follows immediately from the same property for the dimension of vector spaces.

Quotient by a non-zero divisor

Let A be a graded algebra and f a homogeneous element of degree d in A which is not a zero divisor. Then we have

$HS_{A/f}(t)=(1-t^{d})\,HS_{A}(t)\,.$

It follows immediately from the additivity on the exact sequence

$0\;\rightarrow \;A^{[d]}\;{\xrightarrow {f}}\;A\;\rightarrow \;A/f\rightarrow \;0\,,$

where the arrow labeled f is the multiplication by f and $A^{[d]}$ is the graded algebra, which is obtained from A by shifting the degrees by d, in order that the multiplication by f has degree 0. This implies that $HS_{A^{[d]}}(t)=t^{d}\,HS_{A}(t)\,.$

Hilbert series and Hilbert polynomial of a polynomial ring

The Hilbert series of the polynomial ring $R_{n}=K[x_{0},\ldots ,x_{n}]$ is

$HS_{R_{n}}(t)={\frac {1}{(1-t)^{n+1}}}\,.$

It follows that the Hilbert polynomial is

$HP_{R_{n}}(k)={{k+n} \choose {n}}={\frac {(k+1)\cdots (k+n)}{n!}}\,.$

The proof that the Hilbert series has this simple form is obtained by applying recursively the previous formula for the quotient by a non zero divisor (here $x_{n}$) and remarking that $HS_{K}(t)=1\,.$

Shape of the Hilbert series and dimension

A graded algebra A generated by homogeneous elements of degree 1 has Krull dimension zero if the maximal homogeneous ideal, that is the ideal generated by the homogeneous elements of degree 1, is nilpotent. This implies that the dimension of A as a K-vector space is finite and the Hilbert series of A is a polynomial P(t) such that P(1) is equal to the dimension of A as a K vector space.

If the Krull dimension of A is positive, there is a homogeneous element f of degree one which is not a zero divisor (in fact almost all elements of degree one have this property). The Krull dimension of A/f is the Krull dimension of A minus one.

The additivity of Hilbert series shows that $HS_{A/f}(t)=(1-t)\,HS_{A}(t)$. Iterating this a number of times equal to the Krull dimension of A, we get eventually an algebra of dimension 0 whose Hilbert series is a polynomial P(t). This show that the Hilbert series of A is

$HS_{A}(t)={\frac {P(t)}{(1-t)^{d}}}$

where the polynomial P(t) is such that P(1) ≠ 0 and d is the Krull dimension of A.

This formula for the Hilbert series implies that the degree of the Hilbert polynomial is d and that its leading coefficient is P(1)/d!.

Degree of a projective variety and Bézout's theorem

The Hilbert series allows us to compute the degree of an algebraic variety as the value at 1 of the numerator of the Hilbert series. This provides also a simple proof of Bézout's theorem. For this purpose, let us consider an projective algebraic set Template:Mvar defined as the set of the zeros of a homogeneous ideal $I\subset k[x_{0},x_{1},\ldots ,x_{n}]$, where Template:Mvar is a field, and let $R=k[x_{1},\ldots ,x_{n}]/I$ be the ring of the regular functions on the algebraic set (in this section, we do not need that the algebraic set be irreducible nor that the ideal is prime).

If the dimension of Template:Mvar, equal to the dimension of Template:Mvar is Template:Mvar, the degree of Template:Mvar is the number of points of intersection, counted with multiplicity, of Template:Mvar with the intersection of $d$ hyperplanes in general position. This implies that the equations of these hyperplanes, say $h_{1},\ldots ,h_{d},$ are a regular sequence, and that we have the exact sequences

$0\;\rightarrow \;\left(R/\langle h_{1},\ldots ,h_{k-1}\rangle \right)^{[1]}\;{\xrightarrow {h_{k}}}\;R/\langle h_{1},\ldots ,h_{k-1}\rangle \;\rightarrow \;R/\langle h_{1},\ldots ,h_{k}\rangle \;\rightarrow \;0,$

for $k=1,\ldots ,d.$ This implies that

$HS_{R/\langle h_{1},\ldots ,h_{d}\rangle }(t)=(1-t)^{d}\,HS_{R}(t)$

is a polynomial, which is equal to the numerator $P(t)$ of the Hilbert series of Template:Mvar. After dehomogeneizing by putting $x_{0}=1$, Jordan-Hölder theorem for Artinian rings allows to prove that $P(1)$ is the degree of the algebraic set Template:Mvar.

Similarly, if Template:Mvar is a homogeneous polynomial of degree $\delta$, which is not a zero divisor in Template:Mvar, the exact sequence

$0\;\rightarrow \;R^{[\delta ]}\;{\xrightarrow {f}}\;R\;\rightarrow \;R/\langle f\rangle \;\rightarrow \;0,$

shows that

$HS_{R/\langle f\rangle }(t)=(1-t^{\delta })HS_{R}(t).$

Looking on the numerators this proves the following generalization of Bézout's theorem.theorem:

If Template:Mvar is a homogeneous polynomial of degree $\delta$, which is not a zero divisor in Template:Mvar, then the degree of the intersection of Template:Mvar with the hypersurface defined by Template:Mvar is the product of the degree of Template:Mvar by $\delta$ .

The usual Bézout's theorem is easily deduced by starting from a hypersurface and intersecting it, one after the other, with $n-1$ other hypersurfaces.

Computation of Hilbert series and Hilbert polynomial

The Hilbert polynomial is easily deducible from the Hilbert series. This section describes how the Hilbert series may be computed in the case of a quotient of a polynomial ring, filtered or graded by the total degree.

Thus let K a field, $R=K[x_{1},\ldots ,x_{n}]$ be a polynomial ring and I be an ideal in R. Let H be the homogeneous ideal generated by the homogeneous parts of highest degree of the elements of I. If I is homogeneous, then H=I. Finally let B be a Gröbner basis of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the leading monomials of the elements of B.

The computation of the Hilbert series is based on the fact that the filtered algebra R/I and the graded algebras R/H and R/G have the same Hilbert series.

Thus the computation of the Hilbert series is reduced, through the computation of a Gröbner basis, to the same problem for an ideal generated by monomials, which is usually much easier than the computation of the Gröbner basis. The computational complexity of the whole computation depends mainly on the regularity, which is the degree of the numerator of the Hilbert series. In fact the Gröbner basis may be computed by linear algebra over the polynomials of degree bounded by the regularity.

The computation of Hilbert series and Hilbert polynomials are available in most computer algebra systems. For example in both Maple and Magma these functions are named HilbertSeries and HilbertPolynomial.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.