Zariski's lemma

From formulasearchengine
Jump to navigation Jump to search

In mathematics, a linearised polynomial (or q- polynomial) is a polynomial for which the exponents of all the constituent monomials are powers of q and the coefficients come from some extension field of the finite field of order q.

We write a typical example as

This special class of polynomials is important from both a theoretical and an applications viewpoint.[1] The highly structured nature of their roots makes these roots easy to determine.

Properties

  • The map xL(x) is a linear map over any field containing Fq
  • The set of roots of L is an Fq-vector space and is closed under the q-Frobenius map
  • Conversely, if U is any Fq-linear subspace of some finite field containing Fq, then the polynomial that vanishes exactly on U is a linearised polynomial.
  • The set of linearised polynomials over a given field is closed under addition and composition of polynomials.
  • If L is a nonzero linearised polynomial over with all of its roots lying in the field an extension field of , then each root of L has the same multiplicity, which is either 1, or a positive power of q.[2]

Symbolic multiplication

In general, the product of two linearised polynomials will not be a linearized polynomial, but since the composition of two linearised polynomials results in a linearised polynomial, composition may be used as a replacement for multiplication and, for this reason, composition is often called symbolic multiplication in this setting. Notationally, if L1(x) and L2(x) are linearised polynomials we define

when this point of view is being taken.

Associated polynomials

The polynomials L(x) and

are q - associates (note: the exponents "qi " of L(x) have been replaced by "i" in l(x)). More specifically, l(x} is called the conventional q-associate of L(x), and L(x) is the linearised q-associate of l(x).

q-polynomials over Fq

Linearised polynomials with coefficients in Fq have additional properties which make it possible to define symbolic division, symbolic reducibility and symbolic factorizaton. Two important examples of this type of linearised polynomial are the Frobenius automorphism and the trace function .

In this special case it can be shown that, as an operation, symbolic multiplication is commutative, associative and distributes over ordinary addition.[3] Also, in this special case, we can define the operation of symbolic division. If L(x) and L1(x) are linearised polynomials over Fq, we say that L1(x) symbolically divides L(x) if there exists a linearised polynomial L2(x) over Fq for which:

If L1(x) and L2(x) are linearised polynomials over Fq with conventional q-associates l1(x) and l2(x) respectively, then L1(x) symbolically divides L2(x) if and only if l1(x) divides l2(x).[4] Furthermore, L1(x) divides L2(x) in the ordinary sense in this case.[5]

A linearised polynomial L(x) over Fq of degree > 1 is symbolically irreducible over Fq if the only symbolic decompositions

with Li over Fq are those for which one of the factors has degree 1. Note that a symbolically irreducible polynomial is always reducible in the ordinary sense since any linearised polynomial of degree > 1 has the nontrivial factor x. A linearised polynomial L(x) over Fq is symbolically irreducible if and only if its conventional q-associate l(x) is irreducible over Fq.

Every q-polynomial L(x) over Fq of degree > 1 has a symbolic factorization into symbolically irreducible polynomials over Fq and this factorization is essentially unique (up to rearranging factors and multiplying by nonzero elements of Fq.)

For example,[6] consider the 2-polynomial L(x) = x16 + x8 + x2 + x over F2 and its conventional 2-associate l(x) = x4 + x3 + x + 1. The factorization into irreducibles of l(x) = (x2 + x + 1)(x + 1)2 in F2[x], gives the symbolic factorization

Affine polynomials

Let L be a linearised polynomial over . A polynomial of the form is an affine polynomial over .

Theorem: If A is a nonzero affine polynomial over with all of its roots lying in the field an extension field of , then each root of A has the same multiplicity, which is either 1, or a positive power of q.[7]


Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 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