NumberDB$^\beta$ Advanced search Tags Tables Help Login
Fekete polynomials $f_p$
edit · history · discussion · files · short url · polynomial number theory
Polynomials
$p$ 
$f_p(x)$
3:
-x^2 + x
5:
x^4 - x^3 - x^2 + x
7:
-x^6 - x^5 + x^4 - x^3 + x^2 + x
11:
-x^10 + x^9 - x^8 - x^7 - x^6 + x^5 + x^4 + x^3 - x^2 + 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
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
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
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
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
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
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
41:
x^40 + 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
43:
-x^42 + x^41 + x^40 - 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
47:
-x^46 - x^45 - x^44 - x^43 + x^42 - x^41 - x^40 - 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
53:
x^52 - x^51 - x^50 + x^49 - x^48 + x^47 + x^46 - x^45 + x^44 + x^43 + x^42 - x^41 + x^40 - 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
59:
-x^58 + x^57 - x^56 - x^55 - x^54 + x^53 - x^52 + x^51 - x^50 + x^49 + x^48 - x^47 + x^46 + x^45 - x^44 - x^43 - x^42 + x^41 - x^40 - 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
61:
x^60 - x^59 + x^58 + x^57 + x^56 - x^55 - x^54 - x^53 + x^52 - x^51 - x^50 + x^49 + x^48 + x^47 + x^46 + x^45 - x^44 - x^43 + x^42 + x^41 - x^40 + 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
67:
-x^66 + x^65 + x^64 - x^63 + x^62 - x^61 + x^60 + x^59 - x^58 - x^57 + x^56 + x^55 + x^54 - x^53 - x^52 - x^51 - x^50 + x^49 - x^48 + x^47 - x^46 - x^45 - x^44 - x^43 - x^42 - x^41 + x^40 + 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
71:
-x^70 - x^69 - x^68 - x^67 - x^66 - x^65 + x^64 - x^63 - x^62 - x^61 + x^60 - x^59 + x^58 + x^57 - x^56 - x^55 + x^54 - x^53 - x^52 - x^51 + x^50 + x^49 + x^48 - x^47 - x^46 + x^45 - x^44 + x^43 - x^42 - x^41 + x^40 - 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
73:
x^72 + x^71 + x^70 + x^69 - x^68 + x^67 - x^66 + x^65 + x^64 - x^63 - x^62 + x^61 - x^60 - x^59 - x^58 + x^57 - x^56 + x^55 + x^54 - x^53 - x^52 - x^51 + x^50 + x^49 + x^48 - x^47 + x^46 - x^45 - x^44 - x^43 - x^42 + x^41 - x^40 - 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
79:
-x^78 - x^77 + x^76 - x^75 - x^74 + x^73 + x^72 - x^71 - x^70 - x^69 - x^68 + x^67 - x^66 + x^65 + x^64 - x^63 + x^62 - x^61 - x^60 - x^59 - x^58 - x^57 - x^56 + x^55 - x^54 - x^53 + x^52 + x^51 + x^50 + x^49 - x^48 - x^47 + x^46 + x^45 + x^44 - x^43 + x^42 - x^41 + x^40 - 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
83:
-x^82 + x^81 - x^80 - x^79 + x^78 + x^77 - x^76 + x^75 - x^74 - x^73 - x^72 - x^71 + x^70 + x^69 + x^68 - x^67 - x^66 + x^65 + x^64 + x^63 - x^62 + x^61 - x^60 + x^59 - x^58 - x^57 - x^56 - x^55 - x^54 - x^53 - x^52 + x^51 - x^50 + x^49 + x^48 - x^47 - x^46 - x^45 + x^44 - x^43 - x^42 + x^41 + x^40 - 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
89:
x^88 + x^87 - x^86 + x^85 + x^84 - x^83 - x^82 + x^81 + x^80 + x^79 + x^78 - x^77 - x^76 - x^75 - x^74 + x^73 + x^72 + x^71 - x^70 + x^69 + x^68 + x^67 - x^66 - x^65 + x^64 - x^63 - x^62 - x^61 - x^60 - x^59 - x^58 + x^57 - x^56 + x^55 - x^54 + x^53 - x^52 - x^51 + x^50 + x^49 - x^48 + x^47 - x^46 + x^45 + x^44 - x^43 + x^42 - x^41 + x^40 + 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
97:
x^96 + x^95 + x^94 + x^93 - x^92 + x^91 - x^90 + x^89 + x^88 - x^87 + x^86 + x^85 - x^84 - x^83 - x^82 + x^81 - x^80 + x^79 - x^78 - x^77 - x^76 + x^75 - x^74 + x^73 + x^72 - x^71 + x^70 - x^69 - x^68 - x^67 + x^66 + x^65 + x^64 - x^63 + x^62 + x^61 - x^60 - x^59 - x^58 - x^57 - x^56 - x^55 + x^54 + x^53 - x^52 - x^51 + x^50 + x^49 + x^48 + x^47 - x^46 - x^45 + x^44 + x^43 - x^42 - x^41 - x^40 - 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
101:
x^100 - x^99 - x^98 + x^97 + x^96 + x^95 - x^94 - x^93 + x^92 - x^91 - x^90 - x^89 + x^88 + x^87 - x^86 + x^85 + x^84 - x^83 + x^82 + x^81 + x^80 + x^79 + x^78 + x^77 + x^76 - x^75 - x^74 - x^73 - x^72 + x^71 + x^70 - x^69 + x^68 - x^67 - x^66 + x^65 + x^64 - x^63 - x^62 - x^61 - x^60 - x^59 + x^58 - x^57 + x^56 - x^55 + x^54 - x^53 + x^52 - x^51 - x^50 + x^49 - x^48 + x^47 - x^46 + x^45 - x^44 + x^43 - x^42 - x^41 - x^40 - 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
Definition
For an odd prime $p$, the Fekete polynomial is $f_p(x)=\sum_{a=0}^{p-1}\left(\frac{a}{p}\right)x^a$, where $\left(\frac{a}{p}\right)$ is the Legendre symbol [1] [3].
Parameters
$p$
—   odd prime modulus ($p$ is an odd prime)
Formulas
(1)
$f_p(1)=0$, since the nonzero residues and nonresidues modulo $p$ are equally numerous.
Comments
(2)
The natural domain in the literature is $|z|=1$; this table uses $x$ as the variable to match the other polynomial tables in this family.
(3)
The polynomial has degree $p-1$ and constant coefficient $0$. The Legendre sequence of length $p$ instead puts $u_0=1$; that is a different object.
(4)
This table uses $x$ as the variable. The constant term is $0$, and the coefficient of $x^a$ is $1$ for a nonzero quadratic residue modulo $p$ and $-1$ for a quadratic nonresidue modulo $p$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.arith.misc import kronecker_symbol
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing

R = PolynomialRing(ZZ, 'x')
x = R.gen()

def fekete_polynomial(p):
    return sum(ZZ(kronecker_symbol(a, p)) * x**a for a in range(p))

fekete_polynomial(103)      # the next prime after this table
References
[1]
Christian Guenther and Kai-Uwe Schmidt, L^q norms of Fekete and related polynomials, 2016. (arXiv) (doi)
Links
Data properties
Entries are of type: integral polynomial
How well the digits are known: exact
Table is complete: no (it holds every odd prime $p\leq101$)
How they were obtained:

The generator computes the coefficients as Legendre symbols in Sage's integer ring. Before the draft was filled, every entry was checked against Euler's criterion, against the equivalent set of nonzero quadratic residues modulo $p$, and against the specialisation in (1).



API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.