Minimal polynomials of the Pisot numbers less than the golden ratio

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Polynomials

$r$

$m_r(x)$

x^3 - x - 1

comment: $P_{2}(x)=(x-1)\,m_{1}(x)$, and the root of $m_{1}$ in $(1,\varphi)$ is $\theta_{1}$, the plastic ratio and the smallest Pisot number.

x^4 - x^3 - 1

comment: $m_{2}(x)=Q_{2}(x)$, and the root of $m_{2}$ in $(1,\varphi)$ is $\theta_{2}$.

x^5 - x^4 - x^3 + x^2 - 1

comment: $m_{3}(x)=Q_{3}(x)$, and the root of $m_{3}$ in $(1,\varphi)$ is $\theta_{3}$.

x^3 - x^2 - 1

comment: $P_{3}(x)=(x^2-1)\,m_{4}(x)$, and the root of $m_{4}$ in $(1,\varphi)$ is $\theta_{4}$, the supergolden ratio.

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

comment: $m_{5}(x)=Q_{4}(x)$, and the root of $m_{5}$ in $(1,\varphi)$ is $\theta_{5}$.

x^5 - x^3 - x^2 - x - 1

comment: $P_{4}(x)=(x-1)\,m_{6}(x)$, and the root of $m_{6}$ in $(1,\varphi)$ is $\theta_{6}$.

x^7 - x^6 - x^5 + x^2 - 1

comment: $m_{7}(x)=Q_{5}(x)$, and the root of $m_{7}$ in $(1,\varphi)$ is $\theta_{7}$.

x^6 - 2*x^5 + x^4 - x^2 + x - 1

comment: $m_{8}(x)=E(x)$, and the root of $m_{8}$ in $(1,\varphi)$ is $\theta_{8}$.

x^5 - x^4 - x^2 - 1

comment: $P_{5}(x)=(x^2-1)\,m_{9}(x)$, and the root of $m_{9}$ in $(1,\varphi)$ is $\theta_{9}$.

10:

x^8 - x^7 - x^6 + x^2 - 1

comment: $m_{10}(x)=Q_{6}(x)$, and the root of $m_{10}$ in $(1,\varphi)$ is $\theta_{10}$.

11:

x^7 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{6}(x)=(x-1)\,m_{11}(x)$, and the root of $m_{11}$ in $(1,\varphi)$ is $\theta_{11}$.

12:

x^9 - x^8 - x^7 + x^2 - 1

comment: $m_{12}(x)=Q_{7}(x)$, and the root of $m_{12}$ in $(1,\varphi)$ is $\theta_{12}$.

13:

x^7 - x^6 - x^4 - x^2 - 1

comment: $P_{7}(x)=(x^2-1)\,m_{13}(x)$, and the root of $m_{13}$ in $(1,\varphi)$ is $\theta_{13}$.

14:

x^10 - x^9 - x^8 + x^2 - 1

comment: $m_{14}(x)=Q_{8}(x)$, and the root of $m_{14}$ in $(1,\varphi)$ is $\theta_{14}$.

15:

x^9 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{8}(x)=(x-1)\,m_{15}(x)$, and the root of $m_{15}$ in $(1,\varphi)$ is $\theta_{15}$.

16:

x^11 - x^10 - x^9 + x^2 - 1

comment: $m_{16}(x)=Q_{9}(x)$, and the root of $m_{16}$ in $(1,\varphi)$ is $\theta_{16}$.

17:

x^9 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{9}(x)=(x^2-1)\,m_{17}(x)$, and the root of $m_{17}$ in $(1,\varphi)$ is $\theta_{17}$.

18:

x^12 - x^11 - x^10 + x^2 - 1

comment: $m_{18}(x)=Q_{10}(x)$, and the root of $m_{18}$ in $(1,\varphi)$ is $\theta_{18}$.

19:

x^11 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{10}(x)=(x-1)\,m_{19}(x)$, and the root of $m_{19}$ in $(1,\varphi)$ is $\theta_{19}$.

20:

x^13 - x^12 - x^11 + x^2 - 1

comment: $m_{20}(x)=Q_{11}(x)$, and the root of $m_{20}$ in $(1,\varphi)$ is $\theta_{20}$.

21:

x^11 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{11}(x)=(x^2-1)\,m_{21}(x)$, and the root of $m_{21}$ in $(1,\varphi)$ is $\theta_{21}$.

22:

x^14 - x^13 - x^12 + x^2 - 1

comment: $m_{22}(x)=Q_{12}(x)$, and the root of $m_{22}$ in $(1,\varphi)$ is $\theta_{22}$.

23:

x^13 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{12}(x)=(x-1)\,m_{23}(x)$, and the root of $m_{23}$ in $(1,\varphi)$ is $\theta_{23}$.

24:

x^15 - x^14 - x^13 + x^2 - 1

comment: $m_{24}(x)=Q_{13}(x)$, and the root of $m_{24}$ in $(1,\varphi)$ is $\theta_{24}$.

25:

x^13 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{13}(x)=(x^2-1)\,m_{25}(x)$, and the root of $m_{25}$ in $(1,\varphi)$ is $\theta_{25}$.

26:

x^16 - x^15 - x^14 + x^2 - 1

comment: $m_{26}(x)=Q_{14}(x)$, and the root of $m_{26}$ in $(1,\varphi)$ is $\theta_{26}$.

27:

x^15 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{14}(x)=(x-1)\,m_{27}(x)$, and the root of $m_{27}$ in $(1,\varphi)$ is $\theta_{27}$.

28:

x^17 - x^16 - x^15 + x^2 - 1

comment: $m_{28}(x)=Q_{15}(x)$, and the root of $m_{28}$ in $(1,\varphi)$ is $\theta_{28}$.

29:

x^15 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{15}(x)=(x^2-1)\,m_{29}(x)$, and the root of $m_{29}$ in $(1,\varphi)$ is $\theta_{29}$.

30:

x^18 - x^17 - x^16 + x^2 - 1

comment: $m_{30}(x)=Q_{16}(x)$, and the root of $m_{30}$ in $(1,\varphi)$ is $\theta_{30}$.

31:

x^17 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{16}(x)=(x-1)\,m_{31}(x)$, and the root of $m_{31}$ in $(1,\varphi)$ is $\theta_{31}$.

32:

x^19 - x^18 - x^17 + x^2 - 1

comment: $m_{32}(x)=Q_{17}(x)$, and the root of $m_{32}$ in $(1,\varphi)$ is $\theta_{32}$.

33:

x^17 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{17}(x)=(x^2-1)\,m_{33}(x)$, and the root of $m_{33}$ in $(1,\varphi)$ is $\theta_{33}$.

34:

x^20 - x^19 - x^18 + x^2 - 1

comment: $m_{34}(x)=Q_{18}(x)$, and the root of $m_{34}$ in $(1,\varphi)$ is $\theta_{34}$.

35:

x^19 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{18}(x)=(x-1)\,m_{35}(x)$, and the root of $m_{35}$ in $(1,\varphi)$ is $\theta_{35}$.

36:

x^21 - x^20 - x^19 + x^2 - 1

comment: $m_{36}(x)=Q_{19}(x)$, and the root of $m_{36}$ in $(1,\varphi)$ is $\theta_{36}$.

37:

x^19 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{19}(x)=(x^2-1)\,m_{37}(x)$, and the root of $m_{37}$ in $(1,\varphi)$ is $\theta_{37}$.

38:

x^22 - x^21 - x^20 + x^2 - 1

comment: $m_{38}(x)=Q_{20}(x)$, and the root of $m_{38}$ in $(1,\varphi)$ is $\theta_{38}$.

39:

x^21 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{20}(x)=(x-1)\,m_{39}(x)$, and the root of $m_{39}$ in $(1,\varphi)$ is $\theta_{39}$.

40:

x^23 - x^22 - x^21 + x^2 - 1

comment: $m_{40}(x)=Q_{21}(x)$, and the root of $m_{40}$ in $(1,\varphi)$ is $\theta_{40}$.

41:

x^21 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{21}(x)=(x^2-1)\,m_{41}(x)$, and the root of $m_{41}$ in $(1,\varphi)$ is $\theta_{41}$.

42:

x^24 - x^23 - x^22 + x^2 - 1

comment: $m_{42}(x)=Q_{22}(x)$, and the root of $m_{42}$ in $(1,\varphi)$ is $\theta_{42}$.

43:

x^23 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{22}(x)=(x-1)\,m_{43}(x)$, and the root of $m_{43}$ in $(1,\varphi)$ is $\theta_{43}$.

44:

x^25 - x^24 - x^23 + x^2 - 1

comment: $m_{44}(x)=Q_{23}(x)$, and the root of $m_{44}$ in $(1,\varphi)$ is $\theta_{44}$.

45:

x^23 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{23}(x)=(x^2-1)\,m_{45}(x)$, and the root of $m_{45}$ in $(1,\varphi)$ is $\theta_{45}$.

46:

x^26 - x^25 - x^24 + x^2 - 1

comment: $m_{46}(x)=Q_{24}(x)$, and the root of $m_{46}$ in $(1,\varphi)$ is $\theta_{46}$.

47:

x^25 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{24}(x)=(x-1)\,m_{47}(x)$, and the root of $m_{47}$ in $(1,\varphi)$ is $\theta_{47}$.

48:

x^27 - x^26 - x^25 + x^2 - 1

comment: $m_{48}(x)=Q_{25}(x)$, and the root of $m_{48}$ in $(1,\varphi)$ is $\theta_{48}$.

49:

x^25 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{25}(x)=(x^2-1)\,m_{49}(x)$, and the root of $m_{49}$ in $(1,\varphi)$ is $\theta_{49}$.

50:

x^28 - x^27 - x^26 + x^2 - 1

comment: $m_{50}(x)=Q_{26}(x)$, and the root of $m_{50}$ in $(1,\varphi)$ is $\theta_{50}$.

51:

x^27 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{26}(x)=(x-1)\,m_{51}(x)$, and the root of $m_{51}$ in $(1,\varphi)$ is $\theta_{51}$.

52:

x^29 - x^28 - x^27 + x^2 - 1

comment: $m_{52}(x)=Q_{27}(x)$, and the root of $m_{52}$ in $(1,\varphi)$ is $\theta_{52}$.

53:

x^27 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{27}(x)=(x^2-1)\,m_{53}(x)$, and the root of $m_{53}$ in $(1,\varphi)$ is $\theta_{53}$.

54:

x^30 - x^29 - x^28 + x^2 - 1

comment: $m_{54}(x)=Q_{28}(x)$, and the root of $m_{54}$ in $(1,\varphi)$ is $\theta_{54}$.

55:

x^29 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{28}(x)=(x-1)\,m_{55}(x)$, and the root of $m_{55}$ in $(1,\varphi)$ is $\theta_{55}$.

56:

x^31 - x^30 - x^29 + x^2 - 1

comment: $m_{56}(x)=Q_{29}(x)$, and the root of $m_{56}$ in $(1,\varphi)$ is $\theta_{56}$.

57:

x^29 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{29}(x)=(x^2-1)\,m_{57}(x)$, and the root of $m_{57}$ in $(1,\varphi)$ is $\theta_{57}$.

58:

x^32 - x^31 - x^30 + x^2 - 1

comment: $m_{58}(x)=Q_{30}(x)$, and the root of $m_{58}$ in $(1,\varphi)$ is $\theta_{58}$.

59:

x^31 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{30}(x)=(x-1)\,m_{59}(x)$, and the root of $m_{59}$ in $(1,\varphi)$ is $\theta_{59}$.

60:

x^33 - x^32 - x^31 + x^2 - 1

comment: $m_{60}(x)=Q_{31}(x)$, and the root of $m_{60}$ in $(1,\varphi)$ is $\theta_{60}$.

61:

x^31 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{31}(x)=(x^2-1)\,m_{61}(x)$, and the root of $m_{61}$ in $(1,\varphi)$ is $\theta_{61}$.

62:

x^34 - x^33 - x^32 + x^2 - 1

comment: $m_{62}(x)=Q_{32}(x)$, and the root of $m_{62}$ in $(1,\varphi)$ is $\theta_{62}$.

63:

x^33 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{32}(x)=(x-1)\,m_{63}(x)$, and the root of $m_{63}$ in $(1,\varphi)$ is $\theta_{63}$.

64:

x^35 - x^34 - x^33 + x^2 - 1

comment: $m_{64}(x)=Q_{33}(x)$, and the root of $m_{64}$ in $(1,\varphi)$ is $\theta_{64}$.

65:

x^33 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{33}(x)=(x^2-1)\,m_{65}(x)$, and the root of $m_{65}$ in $(1,\varphi)$ is $\theta_{65}$.

66:

x^36 - x^35 - x^34 + x^2 - 1

comment: $m_{66}(x)=Q_{34}(x)$, and the root of $m_{66}$ in $(1,\varphi)$ is $\theta_{66}$.

67:

x^35 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{34}(x)=(x-1)\,m_{67}(x)$, and the root of $m_{67}$ in $(1,\varphi)$ is $\theta_{67}$.

68:

x^37 - x^36 - x^35 + x^2 - 1

comment: $m_{68}(x)=Q_{35}(x)$, and the root of $m_{68}$ in $(1,\varphi)$ is $\theta_{68}$.

69:

x^35 - x^34 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{35}(x)=(x^2-1)\,m_{69}(x)$, and the root of $m_{69}$ in $(1,\varphi)$ is $\theta_{69}$.

70:

x^38 - x^37 - x^36 + x^2 - 1

comment: $m_{70}(x)=Q_{36}(x)$, and the root of $m_{70}$ in $(1,\varphi)$ is $\theta_{70}$.

71:

x^37 - x^35 - x^34 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{36}(x)=(x-1)\,m_{71}(x)$, and the root of $m_{71}$ in $(1,\varphi)$ is $\theta_{71}$.

72:

x^39 - x^38 - x^37 + x^2 - 1

comment: $m_{72}(x)=Q_{37}(x)$, and the root of $m_{72}$ in $(1,\varphi)$ is $\theta_{72}$.

73:

x^37 - x^36 - x^34 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{37}(x)=(x^2-1)\,m_{73}(x)$, and the root of $m_{73}$ in $(1,\varphi)$ is $\theta_{73}$.

74:

x^40 - x^39 - x^38 + x^2 - 1

comment: $m_{74}(x)=Q_{38}(x)$, and the root of $m_{74}$ in $(1,\varphi)$ is $\theta_{74}$.

75:

x^39 - x^37 - x^36 - x^35 - x^34 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{38}(x)=(x-1)\,m_{75}(x)$, and the root of $m_{75}$ in $(1,\varphi)$ is $\theta_{75}$.

76:

x^41 - x^40 - x^39 + x^2 - 1

comment: $m_{76}(x)=Q_{39}(x)$, and the root of $m_{76}$ in $(1,\varphi)$ is $\theta_{76}$.

77:

x^39 - x^38 - x^36 - x^34 - x^32 - x^30 - x^28 - x^26 - x^24 - x^22 - x^20 - x^18 - x^16 - x^14 - x^12 - x^10 - x^8 - x^6 - x^4 - x^2 - 1

comment: $P_{39}(x)=(x^2-1)\,m_{77}(x)$, and the root of $m_{77}$ in $(1,\varphi)$ is $\theta_{77}$.

78:

x^42 - x^41 - x^40 + x^2 - 1

comment: $m_{78}(x)=Q_{40}(x)$, and the root of $m_{78}$ in $(1,\varphi)$ is $\theta_{78}$.

79:

x^41 - x^39 - x^38 - x^37 - x^36 - x^35 - x^34 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28 - x^27 - x^26 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1

comment: $P_{40}(x)=(x-1)\,m_{79}(x)$, and the root of $m_{79}$ in $(1,\varphi)$ is $\theta_{79}$.

Definition

Let $\varphi=(1+\sqrt5)/2$ be the golden ratio, and let $\theta_r$ be the $r$-th smallest Pisot-Vijayaraghavan number [4] in the interval $(1,\varphi)$. The table stores the monic minimal polynomial $m_r(x)\in\mathbb{Z}[x]$ of $\theta_r$.

Parameters

$r$

— rank ($r\geq1$)

Formulas

(1)

The roots $\theta_r$ in $(1,\varphi)$ come from $P_n(x)=x^n(x^2-x-1)+1$ and $Q_n(x)=x^n(x^2-x-1)+x^2-1$ for $n\geq2$, together with $E(x)=x^6-2x^5+x^4-x^2+x-1$.

Comments

(2)

Dufresnoy and Pisot proved that $\varphi$ is the smallest limit point of the set of Pisot numbers and determined all Pisot numbers in $(1,\varphi)$ [1]. Each $m_r(x)$ is the irreducible factor with a root in $(1,\varphi)$ of one of the polynomials $P_n$, $Q_n$ or $E$ of Formula (1) [2] [3].

(3)

The Mahler measure of each polynomial here is $\theta_r$: the polynomial is monic, $\theta_r$ is its only root outside the unit circle, and all other roots lie inside the unit disc.

Programs

(P1)

Sage

R.<x> = QQ[]
P = lambda n: x^n*(x^2 - x - 1) + 1
Q = lambda n: x^n*(x^2 - x - 1) + x^2 - 1
E = x^6 - 2*x^5 + x^4 - x^2 + x - 1
RR = RealIntervalField(400)
phi = (RR(1) + RR(5).sqrt())/2
def pisot_factor(f):
    out = []
    for g, e in f.factor():
        out += [(root, g) for root, m in g.roots(RR)
                if root.lower() > 1 and root.upper() < phi.lower()]
    return out[0]
polys = [P(n) for n in range(2, 30)] + [Q(n) for n in range(2, 30)] + [E]
[g for root, g in sorted(pisot_factor(f) for f in polys)[:10]]

References

[1]

J. Dufresnoy and Ch. Pisot, Etude de certaines fonctions meromorphes bornees sur le cercle unite. Application a un ensemble ferme d'entiers algebriques, Annales Scientifiques de l'Ecole Normale Superieure 72 (1955), 69-92. (doi) (MR)

[2]

M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse and J.-P. Schreiber, Pisot and Salem Numbers, Birkhauser, 1992. (doi)

[3]

James McKee and Chris Smyth, Salem numbers, Pisot numbers, Mahler measure and graphs, Experimental Mathematics 14 (2005), 211-229. (arXiv) (doi)

Links

[4]

Wikipedia: Pisot-Vijayaraghavan number

Similar tables

Pisot numbers less than the golden ratio — stores the root $\theta_r$ in $(1,\varphi)$ of each polynomial $m_r(x)$ held here

Golden ratio — holds the smallest accumulation point of the set of Pisot numbers

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds the minimal polynomials for $\theta_1$ to $\theta_{79}$, matching the range in the root table exactly)

How they were obtained:

The generator forms the exact integer polynomials given in Formula (1) and factors them over $\mathbb Q$. It selects the irreducible factor whose real root lies in $(1,\varphi)$ and stores the monic factor exactly.

A verification run checks that this root is contained in the corresponding stored interval in the root table. The first ten polynomials agree with the source table quoted by Wikipedia's small-Pisot-numbers section.