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33-756: [REDACTED] Look up HP  or hp in Wiktionary, the free dictionary. HP may refer to: Businesses, groups, organisations [ edit ] HP Inc. , an American technology company Hewlett-Packard , the predecessor to HP before the 2015 split Hewlett Packard Enterprise , the other company created as a result of the split HP Foods , British food products company Handley Page , an aircraft company Hindustan Petroleum , Indian petroleum company, subsidiary of Oil and Natural Gas Corporation America West Airlines (1981-2006), an American airline (IATA code HP) Amapola Flyg (2004-present),

66-638: A Swedish airline (IATA code HP) HP Books, an imprint of the Penguin Group Populist Party (Turkey) ( Halkçı Parti ), a political party in Turkey between 1983 and 1985 Brands, products, items [ edit ] Aero Adventure Aventura HP , an ultralight amphibian aircraft China Railways HP , heavy freight train steam locomotive Hilton-Pacey HP (car) , a British 1920s 3-wheeled cyclecar automobile HP Sauce , British sauce named after Houses of Parliament Hy-Tek HP ,

99-413: A fictional character in J.K. Rowling's Wizarding World fictional universe Arts, entertainment, media [ edit ] Harry Potter , a novel series by J.   K. Rowling Hello Project (H!P), a J-pop idol brand under Japanese music company Up-Front Group Horse-Power: Ballet Symphony , a 1932 ballet composed by Carlos Chávez Hot Package , a TV show created by Adult Swim HP: To

132-501: A flat family π : X → S {\displaystyle \pi :X\to S} has the same Hilbert polynomial over any closed point s ∈ S {\displaystyle s\in S} . This is used in the construction of the Hilbert scheme and Quot scheme . Consider a finitely generated graded commutative algebra S over a field K , which

165-498: A machine that transfers thermal energy between spaces Other uses [ edit ] Half-pay , former UK pay for military personnel not on active service Haltepunkt , German for "railway halt" Higher Power , a concept in Alcoholics Anonymous and in similar twelve-step organisations Hire purchase , a type of sales-instalment plan Hit points or health points in video games Hollow-point bullet ,

198-536: 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 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

231-478: A projective algebraic set V , defined as the set of the zeros of a homogeneous ideal I ⊂ k [ x 0 , x 1 , … , x n ] {\displaystyle I\subset k[x_{0},x_{1},\ldots ,x_{n}]} , where k is a field, and let R = k [ x 0 , … , x n ] / I {\displaystyle R=k[x_{0},\ldots ,x_{n}]/I} be

264-570: A regular sequence implies the existence of exact sequences for k = 0 , … , d . {\displaystyle k=0,\ldots ,d.} This implies that where P ( t ) {\displaystyle P(t)} is the numerator of the Hilbert series of R . The ring R 1 = R / ⟨ h 0 , … , h d − 1 ⟩ {\displaystyle R_{1}=R/\langle h_{0},\ldots ,h_{d-1}\rangle } has Krull dimension one, and

297-804: A single-engined high-wing trike aircraft Ilford HP , cubic-grained black-and-white photographic film series, and photographic film prefix Walther HP , handgun Places [ edit ] Harrison Plaza , a shopping mall in the Philippines that closed down in 2019 Heart Peaks , a volcano in Canada Himachal Pradesh , a state in India HP postcode area , UK High Point, North Carolina , US People and characters [ edit ] Hal Porter (1911–1984), Australian author Harry Pearson (audio critic) (1937–2014), American audio critic Harry Potter (character) ,

330-535: A type of round Home page , the main page of a website Howiesons Poort , a stone-age technology period in Africa See also [ edit ] [REDACTED] Search for "hp" , "h-p" , or "hps" on Misplaced Pages. HP Garage , museum of Hewlett-Packard's history All pages with titles beginning with HP All pages with titles containing HP H&P (disambiguation) HPS (disambiguation) Topics referred to by

363-514: A unique polynomial H P S ( n ) {\displaystyle HP_{S}(n)} with rational coefficients which is equal to H F S ( n ) {\displaystyle HF_{S}(n)} for n large enough. This polynomial is the Hilbert polynomial , and has the form The least n 0 such that H P S ( n ) = H F S ( n ) {\displaystyle HP_{S}(n)=HF_{S}(n)} for n ≥ n 0

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396-423: Is P ( 1 ) d ! {\displaystyle {\frac {P(1)}{d!}}} . 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 rather simple proof of Bézout's theorem . For showing the relationship between the degree of a projective algebraic set and the Hilbert series, consider

429-500: Is It follows that the Hilbert polynomial is 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 {\displaystyle x_{n}} ) and remarking that H S K ( t ) = 1 . {\displaystyle HS_{K}(t)=1\,.} A graded algebra A generated by homogeneous elements of degree 1 has Krull dimension zero if

462-543: Is a homogeneous ideal I and this defines an isomorphism of graded algebra between R n / I {\displaystyle 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. Hilbert series and Hilbert polynomial are additive relatively to exact sequences . More precisely, if

495-589: Is a polynomial with integer coefficients, and δ {\displaystyle \delta } is the Krull dimension of S . In this case the series expansion of this rational fraction is where is the binomial coefficient for n > − δ , {\displaystyle n>-\delta ,} and is 0 otherwise. If the coefficient of t n {\displaystyle t^{n}} in H S S ( t ) {\displaystyle HS_{S}(t)}

528-532: Is an affine space that contains V 0 . {\displaystyle V_{0}.} This makes V 0 {\displaystyle V_{0}} an affine algebraic set , which has R 0 = R 1 / ⟨ h d − 1 ⟩ {\displaystyle R_{0}=R_{1}/\langle h_{d}-1\rangle } as its ring of regular functions. The linear polynomial h d − 1 {\displaystyle h_{d}-1}

561-483: Is an exact sequence of graded or filtered modules, then we have and This follows immediately from the same property for the dimension of vector spaces. 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 It follows from the additivity on the exact sequence where the arrow labeled f is the multiplication by f , and A [ d ] {\displaystyle A^{[d]}}

594-448: Is called the Hilbert regularity . It may be lower than deg ⁡ P − δ + 1 {\displaystyle \deg P-\delta +1} . The Hilbert polynomial is a numerical polynomial , since the dimensions are integers, but the polynomial almost never has integer coefficients ( Schenck 2003 , pp. 41). All these definitions may be extended to finitely generated graded modules over S , with

627-521: Is different from Wikidata All article disambiguation pages All disambiguation pages HP">HP The requested page title contains unsupported characters : ">". Return to Main Page . Hilbert polynomial These notions have been extended to filtered algebras , and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes . The typical situations where these notions are used are

660-409: Is finitely generated by elements of positive degree. This means that and that S 0 = K {\displaystyle S_{0}=K} . The Hilbert function maps the integer n to 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 If S

693-411: Is generated by h homogeneous elements of positive degrees d 1 , … , d h {\displaystyle d_{1},\ldots ,d_{h}} , then the sum of the Hilbert series is a rational fraction where Q is a polynomial with integer coefficients. If S is generated by elements of degree 1 then the sum of the Hilbert series may be rewritten as where P

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726-435: 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 H S A / ( f ) ( t ) = ( 1 − t ) H S A ( t ) {\displaystyle HS_{A/(f)}(t)=(1-t)\,HS_{A}(t)} . Iterating this

759-636: Is the graded module which is obtained from A by shifting the degrees by d , in order that the multiplication by f has degree 0. This implies that H S A [ d ] ( t ) = t d H S A ( t ) . {\displaystyle HS_{A^{[d]}}(t)=t^{d}\,HS_{A}(t)\,.} The Hilbert series of the polynomial ring R n = K [ x 1 , … , x n ] {\displaystyle R_{n}=K[x_{1},\ldots ,x_{n}]} in n {\displaystyle n} indeterminates

792-430: Is the number of points of intersection, counted with multiplicities, of V with the intersection of d {\displaystyle d} hyperplanes in general position . This implies the existence, in R , of a regular sequence h 0 , … , h d {\displaystyle h_{0},\ldots ,h_{d}} of d + 1 homogeneous polynomials of degree one. The definition of

825-477: Is the ring of regular functions of a projective algebraic set V 0 {\displaystyle V_{0}} of dimension 0 consisting of a finite number of points, which may be multiple points. As h d {\displaystyle h_{d}} belongs to a regular sequence, none of these points belong to the hyperplane of equation h d = 0. {\displaystyle h_{d}=0.} The complement of this hyperplane

858-473: Is thus For n ≥ i − δ + 1 , {\displaystyle n\geq i-\delta +1,} the term of index i in this sum is a polynomial in n of degree δ − 1 {\displaystyle \delta -1} with leading coefficient a i / ( δ − 1 ) ! . {\displaystyle a_{i}/(\delta -1)!.} This shows that there exists

891-566: The homogeneous coordinate ring of V . 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 g 1 , ..., 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 , … , X n ] {\displaystyle R_{n}=K[X_{1},\ldots ,X_{n}]} onto S . Its kernel

924-640: The Highest Level Na! , a 2005 Philippine TV sitcom "HP" (song) , a 2019 song by Maluma Science, engineering, technology [ edit ] Horsepower , a unit of power Haptoglobin , a protein Hypersensitivity pneumonitis , a respiratory inflammation High precipitation supercell , a thunderstorm classification Hilbert polynomial of a graded algebra Horizontal pitch unit, 0.2 inches, used to specify rack-mounted equipment width H/P, hacking/ phreaking Heat pump ,

957-527: The following: The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space . The 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. In addition, they provide useful invariants for families of algebraic varieties because

990-432: 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

1023-413: The only difference that a factor t 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 is defined as the Hilbert polynomial of

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1056-415: The ring of the regular functions on the algebraic set. In this section, one does not need irreducibility of algebraic sets nor primality of ideals. Also, as Hilbert series are not changed by extending the field of coefficients, the field k is supposed, without loss of generality, to be algebraically closed. The dimension d of V is equal to the Krull dimension minus one of R , and the degree of V

1089-440: The same term [REDACTED] This disambiguation page lists articles associated with the title HP . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=HP&oldid=1251249215 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description

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