Hessenberg matrix

In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1 such that all its Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938. The same condition also occurs in the uniqueness problem for Fourier series. Tirukkannapuram Vijayaraghavan and Raphael Salem continued their study in the 1940s. Salem numbers are a closely related set of numbers.

A characteristic property of PV numbers is that their powers approach integers at an exponential rate. Pisot proved a remarkable converse: if α > 1 is a real number such that the sequence

$\Vert {\alpha }^n\Vert$

measuring the distance from its consecutive powers to the nearest integer is square-summable, or ℓ², then α is a Pisot number (and, in particular, algebraic). Building on this characterization of PV numbers, Salem showed that the set S of all PV numbers is closed. Its minimal element is a cubic irrationality known as the plastic number. Much is known about the accumulation points of S. The smallest of them is the golden ratio.

Definition and properties

An algebraic integer of degree n is a root α of an irreducible monic polynomial P(x) of degree n with integer coefficients, its minimal polynomial. The other roots of P(x) are called the conjugates of α. If α > 1 but all other roots of P(x) are real or complex numbers of absolute value less than 1, so that they lie strictly inside the circle |x| = 1 in the complex plane, then α is called a Pisot number, Pisot–Vijayaraghavan number, or simply PV number. For example, the golden ratio, φ ≈ 1.618, is a real quadratic integer that is greater than 1, while the absolute value of its conjugate, −φ⁻¹ ≈ −0.618, is less than 1. Therefore, φ is a Pisot number. Its minimal polynomial is x² − x − 1.

Elementary properties

• Every positive integer except 1 is a PV number. Conversely, every rational PV number is an integer greater than 1.
• If α is an irrational PV number whose minimal polynomial ends in k then α is greater than |k|. Consequently, all PV numbers that are less than 2 are algebraic units.
• If α is a PV number then so are its powers α^k, for all natural exponents.
• Every real algebraic number field K of degree n contains a PV number of degree n. This number is a field generator. The set of all PV numbers of degree n in K is closed under multiplication.
• Given an upper bound M and degree n, there is only a finite number of PV numbers of degree n that are less than M.
• Every PV number is a Perron number (a real algebraic number greater than one all of whose conjugates have smaller absolute value).

Diophantine properties

The main interest in PV numbers is due to the fact that their powers have a very "biased" distribution (mod 1). If α is a PV number and λ is any algebraic integer in the field of α then the sequence

$\Vert \lambda {\alpha }^n\Vert ,$

where ||x|| denotes the distance from the real number x to the nearest integer, approaches 0 at an exponential rate. In particular, it is a square-summable sequence and its terms converge to 0.

Two converse statements are known: they characterize PV numbers among all real numbers and among the algebraic numbers (but under a weaker diophantine assumption).

• Let α be a real number greater than 1 and there exists a non-zero real number λ such that

$\sum _{n=1}^{\mathrm{\infty }}\Vert \lambda {\alpha }^n{\Vert }^2<\mathrm{\infty }.$

Then α is a Pisot number and λ is an algebraic number in the field of α (Pisot's theorem).

• Let α be an algebraic number greater than 1 and there exists a non-zero real number λ such that

$\Vert \lambda {\alpha }^n\Vert \to 0,n\to \mathrm{\infty }.$

Then α is a Pisot number and λ is an algebraic number in the field of α.

A longstanding Pisot–Vijayaraghavan problem asks whether the assumption that α is algebraic can be dropped from the last statement. If the answer is affirmative, Pisot's numbers would be characterized among all real numbers by the simple convergence of ||λαⁿ|| to 0 for some auxiliary real λ. It is known that there are only countably many numbers α with this property. The problem is to decide whether any of them is transcendental.

Topological properties

The set of all Pisot numbers is denoted S. Since Pisot numbers are algebraic, the set S is countable. Raphael Salem proved that this set is closed: it contains all its limit points. His proof uses a constructive version of the main diophantine property of Pisot numbers: given a Pisot number α, a real number λ can be chosen so that 0 < λ ≤ α and

$\sum _{n=1}^{\mathrm{\infty }}\Vert \lambda {\alpha }^n{\Vert }^2\le 9.$

Thus the ℓ² norm of the sequence ||λαⁿ|| can be bounded by a uniform constant independent of α. In the last step of the proof, Pisot's characterization is invoked to conclude that the limit of a sequence of Pisot numbers is itself a Pisot number.

Closedness of S implies that it has a minimal element. Carl Ludwig Siegel proved that it is the positive root of the equation x³ − x − 1 = 0 (plastic constant) and is isolated in S. He constructed two sequences of Pisot numbers converging to the golden ratio φ from below and asked whether φ is the smallest limit point of S. This was later proved by Dufresnoy and Pisot, who also determined all elements of S that less than φ; not all of them belong to Siegel's two sequences. Vijayaraghavan proved that S has infinitely many limit points; in fact, the sequence of derived sets

$S,S^{\prime },S^{″},\dots$

does not terminate. On the other hand, the intersection $S^{(\omega )}$ of these sets is empty, meaning that the Cantor–Bendixson rank of S is ω. Even more accurately, the order type of S has been determined.

The set of Salem numbers, denoted by T, is intimately related with S. It has been proved that S is contained in the set T' of the limit points of T. It has been conjectured that the union of S and T is closed.

Quadratic irrationals

If $\alpha$ is a quadratic irrational there is only one other conjugate: ${\alpha }^{\prime }$, obtained by changing the sign of the square root in $\alpha$ from

$\alpha =a+\sqrt{D}\text{ to }{\alpha }^{\prime }=a-\sqrt{D}$

or from

$\alpha =\frac{a+\sqrt{D}}{2}\text{ to }{\alpha }^{\prime }=\frac{a-\sqrt{D}}{2}.$

Here a and D are integers and in the second case a is odd and D is congruent to 1 modulo 4.

The required conditions are α > 1 and −1 < α' < 1. These are satisfied in the first case exactly when a > 0 and either $(a-1{)}^2<D<a^2$ or $a^2<D<(a+1{)}^2$. These are satisfied in the second case exactly when $a>0$ and either $(a-2{)}^2<D<a^2$ or $a^2<D<(a+2{)}^2$.

Thus, the first few quadratic irrationals that are PV numbers are:

Value	Root of...	Numerical value
$\frac{1+\sqrt{5}}{2}$	$x^2-x-1$	1.618034... (the golden ratio)
$1+\sqrt{2}$	$x^2-2x-1$	2.414214... (the silver ratio)
$\frac{3+\sqrt{5}}{2}$	$x^2-3x+1$	2.618034...
$1+\sqrt{3}$	$x^2-2x-2$	2.732051... Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.
$\frac{3+\sqrt{13}}{2}$	$x^2-3x-1$	3.302776... Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.
$2+\sqrt{2}$	$x^2-4x+2$	3.414214...
$\frac{3+\sqrt{17}}{2}$	$x^2-3x-2$	3.561553.. Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland..
$2+\sqrt{3}$	$x^2-4x+1$	3.732051... Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.
$\frac{3+\sqrt{21}}{2}$	$x^2-3x-3$	3.791288...Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.
$2+\sqrt{5}$	$x^2-4x-1$	4.236068... Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.

Powers of PV-numbers

Pisot–Vijayaraghavan numbers can be used to generate almost integers: the nth power of a Pisot number approaches integers as n approaches infinity. For example,

$(3+\sqrt{10}{)}^6=27379+8658\sqrt{10}=54757.9999817\dots \approx 54758-\frac{1}{54758}.$

Since $27379$ and $8658\sqrt{10}$ differ by only $0.0000182\dots ,$

$\frac{27379}{8658}=3.162277662\dots$

is extremely close to

$\sqrt{10}=3.162277660\dots .$

Indeed

${\left(\frac{27379}{8658}\right)}^2=10+\frac{1}{8658^2}.$

Higher powers give correspondingly better rational approximations.

This property stems from the fact that for each n, the sum of nth powers of an algebraic integer x and its conjugates is exactly an integer; this follows from an application of Newton's identities. When x is a Pisot number, the nth powers of the other conjugates tend to 0 as n tends to infinity. Since the sum is an integer, the distance from xⁿ to the nearest integer tends to 0 at an exponential rate.

Small Pisot numbers

All Pisot numbers that do not exceed the golden ratio φ have been determined by Dufresnoy and Pisot. The table below lists ten smallest Pisot numbers in the increasing order. The polynomials in this table, with the exception of

$x^6-2x^5+x^4-x^2+x-1,$

are factors of either

$x^n(x^2-x-1)+1$

or

$x^n(x^2-x-1)+(x^2-1).$

The first polynomial is divisible by x² − 1 when n is odd and by x − 1 when n is even. It has one other real zero, which is a PV number. Dividing either polynomial by xⁿ gives expressions that approach x² − x − 1 as n grows very large and have zeros that converge to φ. A complementary pair of polynomials,

$x^n(x^2-x-1)-1$

and

$x^n(x^2-x-1)-(x^2-1)$

yields Pisot numbers that approach φ from above.

	Value	Root of...	Root of...
1	1.3247179572447460260 (plastic number)	$x(x^2-x-1)+(x^2-1)$	$x^3-x-1$
2	1.3802775690976141157 Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.	$x^2(x^2-x-1)+(x^2-1)$	$x^4-x^3-1$
3	1.4432687912703731076 Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.	$x^3(x^2-x-1)+(x^2-1)$	$x^5-x^4-x^3+x^2-1$
4	1.4655712318767680267 Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.	$x^3(x^2-x-1)+1$	$x^3-x^2-1$
5	1.5015948035390873664	$x^4(x^2-x-1)+(x^2-1)$	$x^6-x^5-x^4+x^2-1$
6	1.5341577449142669154	$x^4(x^2-x-1)+1$	$x^5-x^3-x^2-x-1$
7	1.5452156497327552432	$x^5(x^2-x-1)+(x^2-1)$	$x^7-x^6-x^5+x^2-1$
8	1.5617520677202972947	$x^3(x^3-2x^2+x-1)+(x-1)(x^2+1)$	$x^6-2x^5+x^4-x^2+x-1$
9	1.5701473121960543629	$x^5(x^2-x-1)+1$	$x^5-x^4-x^2-1$
10	1.5736789683935169887	$x^6(x^2-x-1)+(x^2-1)$	$x^8-x^7-x^6+x^2-1$

Since these PV numbers are less than 2, they are all units: their minimal polynomials end in 1 or −1.

References

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  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
  
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• 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
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  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
• 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

External links

• Pisot number, Encyclopedia of Mathematics
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