Implement sin_minpoly and tan_minpoly

flatsurf/geometry/subfield.py

@@ -9,8 +9,21 @@
     ISSAC 2015. Proceedings of the 2015 ACM International
     Symposium on Symbolic and Algebraic Computation.
     Pages 259--266.
-    DOI: http://dx.doi.org/10.1145/2755996.2756664
+    DOI: https://doi.org/10.1145/2755996.2756664
+
+.. [Ca2008]
+    Jack S. Calcut
+    The tangent analogues of the Chebyshev polynomials
+
+.. [WaZe1993]
+    William Watkins, Joel Zeitlin
+    The minimal polynomial of cos(2π/n)
+    The American Mathematical Monthly
+    Vol. 100, No. 5 (May, 1993)
+    Pages 471--474.
+    DOI: https://doi.org/10.2307/2324301
 """
+
 from sage.misc.misc_c import prod
 from sage.rings.all import ZZ, QQ, AA, QQbar, NumberField, polygen
 from sage.structure.element import parent
@@ -19,6 +32,7 @@
 from sage.categories.homset import Hom
 from sage.categories.fields import Fields
 from sage.rings.qqbar import do_polred
+from sage.rings.number_field.number_field import CyclotomicField
 
 
 def number_field_elements_from_algebraics(elts, name="a"):
@@ -301,6 +315,29 @@ def chebyshev_T(n, c):
     return T1
 
 
+def tan_chebyshev_T(n, c):
+    r"""
+    Return a pair `(P, Q)`.
+
+    EXAMPLES::
+
+        sage: from flatsurf.geometry.subfield import tan_chebyshev_T
+
+        sage: x = polygen(ZZ)
+        sage: tan_chebyshev_T(3, x)
+        (-x^3 + 3*x, -3*x^2 + 1)
+
+        sage: P, Q = tan_chebyshev_T(3, tan(pi/7))
+        sage: bool(P / Q == tan(3*pi/7))
+        True
+    """
+    P = c
+    Q = parent(c)(1)
+    for i in range(n - 1):
+        P, Q = c * Q + P, Q - c * P
+    return P, Q
+
+
 def cos_minpoly_odd_prime(p, var):
     r"""
     (-1)^k + (-1)^{k-1} T1 + ... + T_k
@@ -352,14 +389,9 @@ def cos_minpoly(n, var="x"):
         x^2 - 2
         sage: cos_minpoly(5)
         x^2 - x - 1
-        sage: cos_minpoly(6)
-        x^2 - 3
-        sage: cos_minpoly(8)
-        x^4 - 4*x^2 + 2
-        sage: cos_minpoly(9)
-        x^3 - 3*x - 1
-        sage: cos_minpoly(10)
-        x^4 - 5*x^2 + 5
+
+        sage: all((2 * QQbar.zeta(2*n)).real().minpoly() == cos_minpoly(n) for n in range(1,25))
+        True
 
         sage: cos_minpoly(90)
         x^24 - 24*x^22 + 252*x^20 - 1519*x^18 + 5796*x^16 - 14553*x^14 + 24206*x^12 - 26169*x^10 + 17523*x^8 - 6623*x^6 + 1182*x^4 - 72*x^2 + 1
@@ -400,3 +432,105 @@ def cos_minpoly(n, var="x"):
         M = M(chebyshev_T(nn, var))
 
     return M
+
+
+def sin_minpoly(n, var="x"):
+    r"""
+    Return the minimal polynomial of 2 cos pi/n
+
+    EXAMPLES::
+
+        sage: from flatsurf.geometry.subfield import sin_minpoly
+
+        sage: sin_minpoly(1)
+        x
+        sage: sin_minpoly(2)
+        x - 2
+        sage: sin_minpoly(3)
+        x^2 - 3
+        sage: sin_minpoly(4)
+        x^2 - 2
+
+        sage: all((2 * QQbar.zeta(2*n)).imag().minpoly() == sin_minpoly(n) for n in range(1,25))
+        True
+    """
+    n = ZZ(n)
+    # use sin(π/n) = cos((n-2)π/(2n))
+    if n % 2:
+        return cos_minpoly(2 * n)
+    elif n % 4 == 2:
+        # n = 4k+2
+        # (n-2)/(2n) = k/(2k+1)
+        p = cos_minpoly(n // 2)
+        if n % 8 == 2:
+            P = p.parent()
+            d = p.degree()
+            return P([(-1) ** (i + d) * c for i, c in enumerate(p)])
+        else:
+            return p
+    else:
+        return cos_minpoly(n)
+
+
+def tan_minpoly(n):
+    r"""
+    Return the minimal polynomial of tan pi/n
+
+    EXAMPLES::
+
+        sage: from flatsurf.geometry.subfield import tan_minpoly
+
+        sage: tan_minpoly(3)
+        x^2 - 3
+        sage: tan_minpoly(4)
+        x - 1
+        sage: tan_minpoly(5)
+        x^4 - 10*x^2 + 5
+        sage: tan_minpoly(6)
+        x^2 - 1/3
+
+        sage: all((QQbar.zeta(2*n).imag() / QQbar.zeta(2*n).real()).minpoly() == tan_minpoly(n) for n in range(3,15))
+        True
+    """
+    N = 2 * n
+    while N % 4:
+        N *= 2
+    K = CyclotomicField(N)
+    z = K.gen() ** (N // (2 * n))
+    I = K.gen() ** (N // 4)
+
+    sin = (z - z.conjugate()) / I
+    cos = z + z.conjugate()
+    return (sin / cos).minpoly()
+
+
+def atan_minpoly(n):
+    r"""
+    Return the minimal polynomial of tan pi/n
+
+    EXAMPLES::
+
+        sage: from flatsurf.geometry.subfield import atan_minpoly
+
+        sage: atan_minpoly(3)
+        x^2 - 1/3
+        sage: atan_minpoly(4)
+        x - 1
+        sage: atan_minpoly(5)
+        x^4 - 2*x^2 + 1/5
+        sage: atan_minpoly(6)
+        x^2 - 3
+
+        sage: all((QQbar.zeta(2*n).real() / QQbar.zeta(2*n).imag()).minpoly() == atan_minpoly(n) for n in range(3,15))
+        True
+    """
+    N = 2 * n
+    while N % 4:
+        N *= 2
+    K = CyclotomicField(N)
+    z = K.gen() ** (N // (2 * n))
+    I = K.gen() ** (N // 4)
+
+    sin = (z - z.conjugate()) / I
+    cos = z + z.conjugate()
+    return (cos / sin).minpoly()