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In mathematics and computer algebra, factorization of polynomials or polynomial factorization refers to factoring a polynomial with coefficients in a given field or in the integers into irreducible factors with coefficients in same domain. Polynomial factorization is one of the fundamental tools of the computer algebra systems.

The history of polynomial factorization starts with Hermann Schubert who in 1793 described the first polynomial factorization algorithm, and Leopold Kronecker, who rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems. In a survey of the subject, Erich Kaltofen wrote in 1982 (see the bibliography, below):

When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient. The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minutes of computer time indicates how successfully this problem has been attacked during the past fifteen years.

Formulation of the question

Polynomial rings over the integers or over a field are unique factorization domains. This means that every element of these rings is a product of a constant and a product of irreducible polynomials (those that are not the product of two non-constant polynomials). Moreover, this decomposition is unique up to multiplication of the factors by invertible constants.

Factorization depends on the base field. For example, the fundamental theorem of algebra, which states that every polynomial with complex coefficients has complex roots, implies that a polynomial with integer coefficients can be factored (with root-finding algorithms) into linear factors over the complex field C. Similarly, over the field of reals, the irreducible factors have degree at most two, while there are polynomials of any degree that are irreducible over the field of rationals Q.

The question of polynomial factorization makes sense only for coefficients in a computable field whose every element may be represented in a computer and for which there are algorithms for the arithmetic operations. Fröhlich and Shepherson have provided examples of such fields for which no factorization algorithm can exist.

The fields of coefficients for which factorization algorithms are known include prime fields (i.e. the field of rationals and prime modular arithmetic) and their finitely generated field extensions. Integer coefficients are also tractable: Kronecker's method is interesting only from a historical point of view, modern algorithms proceed by a succession of:

Square-free factorization
Factorization over finite fields

and reductions:

From the multivariate case to the univariate one
From coefficients in a purely transcendental extension to the multivariate case over the ground field (see below)
From coefficients in an algebraic extension to coefficients in the ground field
From rational coefficients to integer coefficients (see below)
From integer coefficients to coefficients in a prime field with p elements, for a well chosen p.

Primitive part–content factorization

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In this section, we show that factoring over Q (the rational numbers) and over Z (the integers) is essentially the same problem.

The content of a polynomial p ∈ Z[X], denoted "cont(p)", is, up to its sign, the greatest common divisor of its coefficients. The primitive part of p is primpart(p)=p/cont(p), which is a primitive polynomial with integer coefficients. This defines a factorization of p into the product of an integer and a primitive polynomial. This factorization is unique up to the sign of the content. It is a usual convention to choose the sign of the content such that the leading coefficient of the primitive part is positive.

For example,

- 10 x^{2} + 5 x + 5 = (- 5) \cdot (2 x^{2} - x - 1)

is a factorization into content and primitive part.

Every polynomial q with rational coefficients may be written

q = \frac{p}{c},

where p ∈ Z[X] and c ∈ Z: it suffices to take for c a multiple of all denominators of the coefficients of q (for example their product) and p = cq. The content of q is defined as:

cont (q) = \frac{cont (p)}{c},

and the primitive part of q is that of p. As for the polynomials with integer coefficients, this defines a factorization into a rational number and a primitive polynomial with integer coefficients. This factorization is also unique up to the choice of a sign.

For example,

\frac{1}{3} x^{5} + \frac{7}{2} x^{2} + 2 x + 1 = \frac{1}{6} (2 x^{5} + 21 x^{2} + 12 x + 6)

is a factorization into content and primitive part.

Gauss has first proved that the product of two primitive polynomials is also primitive (Gauss's lemma). This implies that a primitive polynomial is irreducible over the rationals if and only if it is irreducible over the integers. This implies also that the factorization over the rationals of a polynomial with rational coefficients is the same as the factorization over the integers of its primitive part. On the other hand, the factorization over the integers of a polynomial with integer coefficients is the product of the factorization of its primitive part by the factorization of its content.

In other words, integer GDD computation allows to reduce the factorization of a polynomial over the rationals to the factorization of a primitive polynomial with integer coefficients, and to reduce the factorization over the integers to the factorization of an integer and a primitive polynomial.

Everything that precedes remains true if Z is replaced by a polynomial ring over a field F and Q is replaced by a field of rational functions over F in the same variables, with the only difference that "up to a sign" must be replaced by "up to the multiplication by an invertible constant in F". This allows to reduce the factorization over a purely transcendental field extension of F to the factorization of multivariate polynomials over F.

Square-free factorization

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If two or more factors of a polynomial are identical to each other, then the polynomial is a multiple of the square of this factor. In the case of univariate polynomials, this results in multiple roots. In this case, then the multiple factor is also a factor of the polynomial's derivative (with respect to any of the variables, if several), which itself is a polynomial of lower degree. In the case of univariate polynomials over the rationals (or more generally over a field of characteristic zero), Yun's algorithm exploits this remark to factorize efficiently the polynomial into factors that are not multiple of a square and are therefore called square-free. To factorize the initial polynomial, it suffices to factorize each square-free factors. This algorithm is therefore the first step of almost all polynomial factorization algorithms.

Yun's algorithm extends easily to the multivariate case by considering a multivariate polynomial as an univariate polynomial over a polynomial ring.

In the case of a polynomial over a finite field, Yun's algorithm applies only if the degree is smaller than the characteristic, because, otherwise, the derivative of a non zero polynomial may be zero (over the field with p elements, the derivative of a polynomial in x^p is always zero). Nevertheless a succession of GCD computations, starting from the polynomial and its derivative, allows to compute the square-free decomposition; see Polynomial factorization over finite fields#Square-free factorization.

Most factorization algorithms, including all the most efficient ones, begin by a square-free factorization.

Classical methods

This section describes textbook methods that can be convenient when computing by hand. These methods are not used for computer computations because they use integer factorization, which at the moment has a much higher complexity than polynomial factorization.

Obtaining linear factors

All linear factors with rational coefficients can be found using the rational root test. If the polynomial to be factored is $a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$ , then all possible linear factors are of the form $b_{1} x - b_{0}$ , where $b_{1}$ is an integer factor of $a_{n}$ and $b_{0}$ is an integer factor of $a_{0}$ . All possible combinations of integer factors can be tested for validity, and each valid one can be factored out using polynomial long division. If the original polynomial is the product of factors, at least two of which are of degree 2 or higher, this technique only provides a partial factorization; otherwise the factorization is complete. Note that in the case of a cubic polynomial, if the cubic is factorisable at all, the rational root test gives a complete factorization, either into a linear factor and an irreducible quadratic factor, or into three linear factors.

Kronecker's method

Since integer polynomials must factor into integer polynomial factors, and evaluating integer polynomials at integer values must produce integers, the integer values of a polynomial can be factored in only a finite number of ways, and produce only a finite number of possible polynomial factors.

For example, consider

f (x) = x^{5} + x^{4} + x^{2} + x + 2

.

If this polynomial factors over Z, then at least one of its factors must be of degree two or less. We need three values to uniquely fit a second degree polynomial. We'll use $f (0) = 2$ , $f (1) = 6$ and $f (- 1) = 2$ . Note that if one of those values were 0 then you already found a root (and so a factor). If none is 0, then each one has a finite amount of divisors. Now, 2 can only factor as

1×2, 2×1, (−1)×(−2), or (−2)×(−1).

Therefore, if a second degree integer polynomial factor exists, it must take one of the values

1, 2, −1, or −2

at $x = 0$ , and likewise at $x = - 1$ . There are eight different ways to factor 6 (one for each divisor of 6), so there are

4×4×8 = 128

possible combinations, of which half can be discarded as the negatives of the other half, corresponding to 64 possible second degree integer polynomials that must be checked. These are the only possible integer polynomial factors of $f (x)$ . Testing them exhaustively reveals that

p (x) = x^{2} + x + 1

constructed from $p (0) = 1$ , $p (1) = 3$ and $p (- 1) = 1$ , factors $f (x)$ .

Dividing $f$ by $p$ gives the other factor $q (x) = x^{3} - x + 2$ , so that $f = p q$ . Now one can test recursively to find factors of $p$ and $q$ . It turns out they both are irreducible over the integers, so that the irreducible factorization of $f$ is

f (x) = p (x) q (x) = (x^{2} + x + 1) (x^{3} - x + 2)

(Van der Waerden, Sections 5.4 and 5.6)

Modern methods

Lifting modular factorization to factorization over the integers

Template:Expand section

Lifting univariate factorization to multivariate factorization

Template:Expand section

Factoring over finite fields

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Uses of LLL algorithm

The first polynomial time algorithm for factoring rational polynomials has been discovered by Lenstra, Lenstra and Lovász and is an application of Lenstra–Lenstra–Lovász lattice basis reduction algorithm, usually called "LLL algorithm". Template:Harv Although theoretically faster in the worst case, their factorization algorithm is not efficient in practice and is not used on computers.

However LLL algorithm is used by the fastest factorization algorithm to lift a modular factorization to a factorization over the integers.

One variation of LLL factorization algorithm runs as follows: calculate a complex (or p-adic) root α of the polynomial P to high precision, then use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm to find an approximate linear relation between 1, α, α², α³, ... with integer coefficients, which with luck, is an exact linear relation and a polynomial factor of P. One can determine a bound for the precision that guarantees that this method produces either a factor, or an irreducibility proof.

Factoring over algebraic extensions (Trager's method)

We can factor a polynomial $p (x) \in K [x]$ , where $K$ is a finite field extension of $ℚ$ . First, using square-free factorization, we may suppose that the polynomial is square-free. Next we write $L = K [x] / p (x)$ explicitly as an algebra over $ℚ$ . We next pick a random element $α \in L$ . By the primitive element theorem, $α$ generates $L$ over $ℚ$ with high probability. If this is the case, we can compute the minimal polynomial, $q (y) \in ℚ [y]$ of $α$ over $ℚ$ . Factoring

q (y) = \prod_{i = 1}^{n} q_{i} (y)

over $ℚ [y]$ , we determine that

L = ℚ [α] = ℚ [y] / q (y) = \prod_{i = 1}^{n} ℚ [y] / q_{i} (y)

(notice that $L$ is a reduced ring since $p (x)$ is square-free), where $α$ corresponds to the element $(y, y, \dots, y)$ . Note that this is the unique decomposition of $L$ as a product fields. Hence this decomposition is the same as

\prod_{i = 1}^{m} K [x] / p_{i} (x)

where

p (x) = \prod_{i = 1}^{m} p_{i} (x)

is the factorization of $p (x)$ over $K [x]$ . By writing $x \in L$ and generators of $K$ as a polynomials in $α$ , we can determine the embeddings of $x$ and $K$ into the components $ℚ [y] / q_{i} (y) = K [x] / p_{i} (x)$ . By finding the minimal polynomial of $x$ in this ring, we have computed $p_{i} (x)$ , and thus factored $p (x)$ over $K .$

Bibliography

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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
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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
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
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
Van der Waerden, Algebra (1970), trans. Blum and Schulenberger, Frederick Ungar.

Normalized Difference Vegetation Index

Contents

Formulation of the question

Primitive part–content factorization

Square-free factorization

Classical methods

Obtaining linear factors

Kronecker's method

Modern methods

Lifting modular factorization to factorization over the integers

Lifting univariate factorization to multivariate factorization

Factoring over finite fields

Uses of LLL algorithm

Factoring over algebraic extensions (Trager's method)

Bibliography

Further reading

Navigation menu

Normalized Difference Vegetation Index

Formulation of the question

Primitive part–content factorization

Square-free factorization

Classical methods

Obtaining linear factors

Kronecker's method

Modern methods

Lifting modular factorization to factorization over the integers

Lifting univariate factorization to multivariate factorization

Factoring over finite fields

Uses of LLL algorithm

Factoring over algebraic extensions (Trager's method)

Bibliography

Further reading

Navigation menu

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