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

$\left(\begin{array}{cccc}0& 1& 2& 3\\ 0& 3& 2& 1\end{array}\right)$.

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

$\left(\begin{array}{cccccccc}0& 1& 2& 3& 4& 5& 6& 7\\ 0& 3& 2& 5& 4& 7& 6& 1\end{array}\right)$.

Rings Z/p^kZ

Consider $g(x)=a{x}^2+bx+c$ 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 $a=0modp$ and $b\ne 0modp$.

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 $g(x)=a_0+\sum _{0<i\le M}{a}_i{x}^i$ defines a permutation for the ring Z/nZ if and only if all the polynomials $g_{p_t}(x)=a_{0,p_t}+\sum _{0<i\le M}{a}_{i,p_t}{x}^i$ 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 $a{x}^2+bx$, such that $a=0mod{p}_1$, but $a\ne 0mod{p}_1^{k_1}$; assume that $a=0mod{p}_i^{k_i}$,i>1. And assume that $b\ne 0mod{p}_i$ 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

$\left(\begin{array}{cccccccccc}0& 1& 2& 3& 4& 5& 6& 7& 8& ...\\ 0& 7& 2& 9& 4& 11& 6& 1& 8& ...\end{array}\right)$.

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 $g^{\prime }(x)\ne 0modp$ 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 $g(x)=a{x}^3+bx$ defines a permutation if and only if for all $y\in Z/pZ$ it is true that $y^2\ne -b/a$, 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

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

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

$\mathrm{d}\mathrm{e}\mathrm{t}\left({\alpha }_{i-j}^{q^j}\right)\mathrm{\ne }\mathrm{0}\mathrm{(}\mathrm{i},\mathrm{j}=\mathrm{0},\mathrm{1},\mathrm{\dots },\mathrm{r}\mathrm{-}\mathrm{1}\mathrm{)}.$

• 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

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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

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References

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  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

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