In today's blog, I will present Newton's formula and show how it leads to Newton's identities. In a future blog, I will provide the proof for the formula.
Definition 1: Newton's Formula for Newton's Identities
For a polynomial of degree n, with roots { r1, ..., rn }:
For any integer i, let si be the sum based on the roots:
si = r1i + r2i + ... + rni.
For any integer j, Let σj be the elementary symmetric polynomial for j in n [see here for details if needed on elementary symmetric polynomials]
Then, Newton's formula is:
where σ0 = 1 and if k is greater than n, σk=0.
We can now use it to build some equations for si and σj.
If k=1, then we have:
s1 = σ1
If k ≥ 2, then we have:
sk = ∑ (i=1, k-1) (-1)i+1sk-iσi + (-1)k+1kσk
Using the above formula gives us:
s1 = σ1
s2 = s1σ1 - 2σ2
s3 = s2σ1 - s1σ2 + 3σ3
s4 = s3σ1 - s2σ2 + s1σ3 - 4σ4
s5 = s4σ1 - s3σ2 + s2σ3 - s1σ4 + 5σ5
...
Now, building each formula based on the previous formula gives us the following formulas for sk in term so of σi:
s1 = σ1
s2 = s1σ1 - 2σ2 = (σ1)σ1 - 2σ2 = σ12 - 2σ2
s3 = s2σ1 - s1σ2 + 3σ3 = (σ12 - 2σ2)σ1 - (σ1)σ2 + 3σ3 = σ13 -3σ1σ2 + 3σ3
s4 = s3σ1 - s2σ2 + s1σ3 - 4σ4 = σ1( σ13 -3σ1σ2 + 3σ3) - σ2(σ12 - 2σ2) + σ3(σ1) - 4σ4 =
= σ14 - 3σ12σ2 + 3σ1σ3 - σ12σ2 + 2σ22 + σ1σ3 - 4σ4 =
= σ14 -4σ12σ2 + 4σ1σ3 + 2σ22 - 4σ4
...
Further, we can use these same formulas for each σi so that:
σ1 = s1
σ2 = (1/2)[s1σ1 - s2]
σ3 = (1/3)[s3 - s2σ1 + s1σ2]
σ4 = (1/4)[-s4 + s3σ1 - s2σ2 + s1σ1
σ5 = (1/5)[s5 - s4σ1 + s3σ2 - s2σ3 + s1σ4]
Now, I will show these equations can be used to derive Newton's identities for cubic polynomials (that is, where n = 3). [See here for review of Newton's identities] where I am assuming an equation of the following form:
x3 + bx2 + cx + d = 0
Here are the justifications for each formula presented previously.
Identity 1: r + s + t = -b
Proof:
σ1 = r + s + t [See Definition 1, here]
σ1 = (-1)1(b) = -b [See Lemma 1, here]
QED
Identity 2: r2 + s2 + t2 = b2 - 2c
Proof:
s2 = r2 + s2 + t2 [See Definition 1 above]
s2 = σ12 - 2σ2 [See formula above]
σ12 - 2σ2 = ((-1)1b)2 - 2(-1)2(c) = b2 - 2c.
QED
Identity 3: r3 + s3 + t3 = -b3 + 3bc - 3d
Proof:
s3 = r3 + s3 + t3 [See Definition 1 above]
s3 = σ13 -3σ1σ2 + 3σ3 [See formula above]
σ13 -3σ1σ2 + 3σ3 = [(-1)1b]3 - 3(-1)1b(-1)2c + 3(-1)3d =
= -b3 +3bc -3d.
QED
Identity 4: rs + rt + st = c
Proof:
σ2 = rs + rt + sr + st +tr + ts [See Definition 1, here]
σ2 = (-1)2c = c
QED
Identity 5: r2s + r2t + s2r + s2t + t2r + t2s = -bc + 3d
Proof:
(rs + rt + sr + st +tr + ts)(r + s + t) - 3rst = r2s + r2t + s2r + s2t + t2r + t2s
(rs + rt + sr + st +tr + ts)(r + s + t) - 3rst = σ2σ1 - 3σ3 = (-1)1b(-1)2c - 3*(-1)3d = -bc + 3d.
QED
Identity 6: r3s + r3t + s3r + s3t + t3r + t3s = b2c - 2c2 - bd
Proof:
(r3 + s3 + t3)(r + s + t) - (r4 + s4 + t4) = r3s + r3t + s3r + s3t + t3r + t3s
(r3 + s3 + t3)(r + s + t) - (r4 + s4 + t4) =( s3)(σ1) - s4
( s3)(σ1) - s4 = (σ13 -3σ1σ2 + 3σ3)(σ1) - (σ14 - 4σ12σ2 + 4σ1σ3 + 2σ22 - 4σ4) =
= σ12σ2 - σ1σ3 - 2σ22 + 4σ4 = [(-1)b]2(-1)2c - 2[(-1)2c]2 - [(-1)b(-1)3d] =
= b2c - 2c2 - bd + 0 = b2c - 2c2 - bd
QED
Identity 7: r2s2 + r2t2 + s2t2 = c2 - 2bd
Proof:
r2s2 + r2t2 + s2t2 = (1/2)(r2 + s2 + t2)(r2 + s2 + t2) - (1/2)(r4 + s4 + t4) - =
= (1/2)s2*s2 - (1/2)s4 =
= (1/2)(σ12 - 2σ2)(σ12 - 2σ2) - (1/2)( σ14 - 4σ12σ2 + 4σ1σ3 + 2σ22 - 4σ4) =
= (1/2)σ14 - 2σ12σ2 + 2σ22 - (1/2)σ14 + 2σ12σ2 - 2σ1σ3 - σ22 + 2σ4 =
= σ22 - 2σ1σ3 + 2σ4 =
= (c)2 - 2*(-1)(b)(-1)(d) + 2*(0) = c2 -2bd.
QED
Identity 8: r3s2 + r3t2 + s3r2 + s3t2 + t3r2 + t3s2 = -bc2 + 2b2d + cd
Proof:
(1) (r2s2 + r2t2 + s2t2)(r + s + t) - (rst)(rs + rt + st) =
r3s2 + r3t2 + s3r2 + s3t2 + t3r2 + t3s2
(2) (r2s2 + r2t2 + s2t2)(r + s + t) - (rst)(rs + rt + st) =
=(σ22 - 2σ1σ3 + 2σ4 )(σ1) - (σ3)(σ2) =
= σ1σ22 - 2σ12σ3 + 2σ1σ4 - σ2σ3 =
= (-1)b(c)2 - 2(b)2(-d) + 2(-b)(0) - (c)(-d) =
= -bc2 + 2b2d + cd
QED
Identity 9: r3s3 + r3t3 + s3t3 = c3 - 3bcd + 3d2
Proof:
(1) r3s3 + r3t3 + s3t3 = (r2s2 + r2t2 + s2t2)(rs + rt + st) - (r3s2t + r3st2 + s3r2t + s3t2r + t3r2s + t3s2r )
(2) (r2s2 + r2t2 + s2t2)(rs + rt + st) - (r3s2t + r3st2 + s3r2t + s3t2r + t3r2s + t3s2r ) = (c2 - 2bd)(σ2) - (bcd - 3d2) =
= (c2 - 2bd)(c) - (bcd - 3d2) = c3 - 2bcd -bcd + 3d2 =
= c3 - 3bcd +3d2
QED
Identity 10: rst = -d
Proof:
σ3 = rst
σ3 = (-1)3d = -d
QED
Identity 11: r2st + s2rt + t2rs = bd
Proof:
r2st + s2rt + t2rs = rst(r + s + t) = σ3*σ1 = (-1)(d)(-1)(b) = bd
QED
Identity 12: r3st + s3rt + t3rs = -b2d + 2cd
Proof:
(rst)(r2 + s2 + t2) = r3st + s3rt + t3rs
(rst)(r2 + s2 + t2) = σ3s2 =
= σ3(σ12 - 2σ2) = (-d)(b2 -2c) = -b2d + 2cd.
QED
Identity 13: r2s2t + r2st2 + rs2t2 = -cd
Proof:
(rst)(rt + rs + st) = r2s2t + r2st2 + rs2t2
(rst)(rt + rs + st) = σ3σ2 = (-d)(c) = -cd.
QED
Identity 14: r3s2t + r3st2 + s3r2t + s3t2r + t3r2s + t3s2r = bcd - 3d2
Proof:
(rst)[(rs + rt + st)(r + s + t) - 3rst] = r3s2t + r3st2 + s3r2t + s3t2r + t3r2s + t3s2r
(rst)[(rs + rt + st)(r + s + t) - 3rst] = σ3[(σ2)(σ1) - 3σ3] =
σ3[(σ2)(σ1) - 3σ3] = (-1)d[c(-b) - 3(-1)d] = (-d)[-bc + 3d] = bcd - 3d2
QED
Identity 15: r3s3t + r3t3s + s3t3r = -c2d + 2bd2
Proof:
(rst)[(1/2)(r2 + s2 + t2)(r2 + s2 + t2) - (1/2)(r4 + s4 + t4) ] = r3s3t + r3t3s + s3t3r
(rst)[(1/2)(r2 + s2 + t2)(r2 + s2 + t2) - (1/2)(r4 + s4 + t4) ] =
= σ3[(1/2)(s2)(s2) - (1/2)s4 ] =
= σ3[(1/2)(σ12 - 2σ2)(σ12 - 2σ2) - (1/2)( σ14 - 4σ12σ2 + 4σ1σ3 + 2σ22 - 4σ4)] =
= σ3[(1/2)σ14 - 2σ12σ2 + 2σ22 - (1/2)σ14 + 2σ12σ2 - 2σ1σ3 - 2σ22 + 2*0)] =
= σ3[σ22 -2σ1σ3] = (-1)d[(c)2 - 2*(-1)b(-1)d] =
= (-d)[c2 - 2bd] = -c2d + bd2.
QED
Identity 16: r2s2t2 = d2
Proof:
r2s2t2 = (rst)2 = (σ3)2 = [(-1)3d]2 = d2
QED
Identity 17: r3s2t2 + s3r2t2 + t3r2s2 = -bd2
Proof:
(rst)(rst)(r + s + t) = r3s2t2 + s3r2t2 + t3r2s2
(rst)(rst)(r + s + t) = σ32*σ1 = (-b)d2= -bd2
QED
Identity 18: r3s3t2 + r3t3s2 + s3t3r2 = cd2
Proof:
(rs + rt + st)(rst)(rst) = r3s3t2 + r3t3s2 + s3t3r2
(rs + rt + st)(rst)(rst) = σ2*σ32 = (c)(d)2 = cd2
QED
Identity 19: r3s3t3 = -d3
Proof:
r3s3t3 = (rst)3 = (σ3)3 = (-d)3 = -d3
QED
References
- "Newton's Identities", Wikipedia
- Jean-Pierre Tignol, Galois' Theory of Algebraic Equations
, World Scientific, 2001
- Harold M. Edwards, Galois Theory
, Springer, 1984.