Log-spectral distance

In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map $x\mapsto g(x)$ is a bijection. In case the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. There are some situations in which these are essentially the only permutation polynomials over finite fields.

In the case of finite rings Z/nZ, such polynomials have also been studied and applied in the interleaver component of error detection and correction algorithms.

Quadratic permutation polynomials (QPP)

For the finite ring Z/nZ one can construct quadratic permutation polynomials. Actually it is possible if and only if n is divisible by p² for some prime number p. The construction is surprisingly simple, nevertheless it can produce permutations with certain good properties. That is why it has been used in the interleaver component of turbo codes in 3GPP Long Term Evolution mobile telecommunication standard (see 3GPP technical specification 36.212 e.g. page 14 in version 8.8.0).

Simple examples

Consider $g(x)=2x^{2}+x$ for the ring Z/4Z. One sees: $g(0)=0;~g(1)=3;~g(2)=2;~g(3)=1$, so the polynomial defines the permutation

${\begin{pmatrix}0&1&2&3\\0&3&2&1\end{pmatrix}}$.

Let us consider the same polynomial $g(x)=2x^{2}+x$ for the other ring Z/8Z. One sees: $g(0)=0;~g(1)=3;~g(2)=2;~g(3)=5;~g(4)=4;~g(5)=7;~g(6)=6;~g(7)=1;$, so the polynomial defines the permutation

${\begin{pmatrix}0&1&2&3&4&5&6&7\\0&3&2&5&4&7&6&1\end{pmatrix}}$.

Rings Z/p^kZ

Consider $g(x)=ax^{2}+bx+c$ for the ring Z/p^kZ.

Lemma: for k=1 (i.e. Z/pZ) such polynomial defines a permutation only in the case a=0 and b not equal to zero. So the polynomial is not quadratic, but linear.

Lemma: for k>1 ( Z/p^kZ) such polynomial defines a permutation if and only if $a=0~mod~p$ and $b\neq 0~mod~p$.

Rings Z/nZ

Consider $n=p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{l}^{k_{l}}$, where p_t are prime numbers.

Lemma: any polynomial $g(x)=a_{0}+\sum _{0<i\leq M}a_{i}x^{i}$ defines a permutation for the ring Z/nZ if and only if all the polynomials $g_{p_{t}}(x)=a_{0,p_{t}}+\sum _{0<i\leq M}a_{i,p_{t}}x^{i}$ defines the permutations for all rings $Z/p_{t}^{k_{t}}Z$, where $a_{j,p_{t}}$ are remainders of $a_{j}$ modulo $p_{t}^{k_{t}}$.

As a corollary one can construct plenty quadratic permutation polynomials using the following simple construction. Consider $n=p_{1}^{k_{1}}p_{2}^{k_{2}}...p_{l}^{k_{l}}$, assume that k₁ >1.

Consider $ax^{2}+bx$, such that $a=0~mod~p_{1}$, but $a\neq 0~mod~p_{1}^{k_{1}}$; assume that $a=0~mod~p_{i}^{k_{i}}$,i>1. And assume that $b\neq 0~mod~p_{i}$ for all i=1...l. (For example one can take $a=p_{1}p_{2}^{k_{2}}...p_{l}^{k_{l}}$ and $b=1$). Then such polynomial defines a permutation.

To see this we observe that for all primes p_i,i>1, the reduction of this quadratic polynomial modulo p_i is actually linear polynomial and hence is permutation by trivial reason. For the first prime number we should use the lemma discussed previously to see that it defines the permutation.

For example, consider Z/12Z and polynomial $6x^{2}+x$. It defines a permutation

${\begin{pmatrix}0&1&2&3&4&5&6&7&8&...\\0&7&2&9&4&11&6&1&8&...\end{pmatrix}}$.

Higher degree polynomials

A polynomial g(x) for the ring Z/p^kZ is a permutation polynomial if and only if it permutes the finite field Z/pZ and $g'(x)\neq 0~mod~p$ for all x in Z/p^kZ, where g′(x) is the formal derivative of g(x).

Lemma: consider the finite field Z/pZ for some prime number p. The cubic polynomial $g(x)=ax^{3}+bx$ defines a permutation if and only if for all $y\in Z/pZ$ it is true that $y^{2}\neq -b/a$, i.e. the Legendre symbol $\left({\frac {-b/a}{p}}\right)=-1.$.

Evaluation of the Legendre symbol can be achieved with the help of quadratic reciprocity law.

Finite fields

There are many open questions concerning permutation polynomials defined over finite fields (see Template:Harvtxt and Template:Harvtxt).

If f(x) is a permutation polynomial defined over the finite field GF(q), where q = p^e for some prime p, then so is g(x) = a f(x + b) + c for all a ≠ 0, b and c in GF(q). We say that g(x) is in normalized form if a, b and c are chosen so that g(x) is monic, g(0) = 0 and (provided the characteristic p does not divide the degree n of the polynomial) the coefficient of x^n-1 is 0.

The following list, while not exhaustive, contains almost all of the known major classes of permutation polynomials over finite fields.

• Template:Harvtxt lists all normalized permutation polynomials of degree at most 5.
• x^k permutes GF(q) if and only if (k, q - 1) = 1.
• If a is in GF(q) then the Dickson polynomial g_k(x,a) permutes GF(q) if and only if (k, q² - 1) = 1.
• If GF(q^r) is an extension of GF(q) of degree r, then

$L(x)=\Sigma _{s=0}^{r-1}\alpha _{s}x^{q^{s}}$

with α_s in GF(q^r) is a linear operator on GF(q^r) over GF(q) and it permutes GF(q^r) if and only if

${\rm {{det}\left(\alpha _{i-j}^{q^{j}}\right)\neq 0\quad (i,j=0,1,\ldots ,r-1).}}$

• If r > 1 is relatively prime to q - 1, s divides q - 1 and g(x^s) has no nonzero root in GF(q) where g(x) is in the polynomial ring GF(q)[x], then x^r(g(x^s))^{(q - 1)/s} permutes GF(q).
• Only a few other specific classes of permutation polynomials over GF(q) have been characterized. Two of these, for example, are:

$x^{(q+m-1)/m}+ax$

where m divides q - 1, and

$x^{r}(x^{d}-a)^{(p^{n}-1)/d}$

where d divides pⁿ - 1.

Geometric examples

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

In finite geometry coordinate descriptions of certain point sets can provide examples of permutation polynomials of higher degree. In particular, the points forming an oval in a finite Projective plane, PG(2,q) with q a power of 2, can be coordinatized in such a way that the relationship between the coordinates is given by an o-polynomial, which is a special type of permutation polynomial over the finite field GF(q).

Schur's conjecture

Let K be an algebraic number field with R the ring of integers. The term "Schur's conjecture" refers to the assertion that, if a polynomial f defined over K is a permutation polynomial on R/P for infinitely many prime ideals P, then f is the composition of Dickson polynomials, degree-one polynomials, and polynomials of the form x^k. In fact, Schur did not make any conjecture in this direction. The notion that he did is due to Fried, who gave a flawed proof of a false version of the result. Correct proofs have been given by Turnwald and Müller.

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
• 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
• 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

Further reading

• 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 Chapter 7.