CHAPTER 1
Polynomials i n On e Variabl e
The study of systems of polynomial equations in many variables requires a good
understanding o f what ca n be sai d abou t on e polynomial equatio n i n one variable .
The purpos e o f thi s chapte r i s t o provid e som e basi c tool s fo r thi s problem . W e
shall conside r th e proble m o f ho w t o comput e an d ho w t o represen t th e zero s o f a
general polynomia l o f degree d in one variable x:
(1.1) p(x) = a
dxd
+ a
d-ixd~l
H ( -
a2x2
+ a xx - f a 0.
1.1. Th e Fundamenta l Theore m o f Algebr a
We begi n b y assumin g tha t th e coefficient s a $ li e i n th e field Q o f rationa l
numbers, wit h a ^ / 0 , wher e th e variabl e x range s ove r th e field C o f comple x
numbers. Ou r startin g poin t i s the fac t tha t C i s algebraically closed .
THEOREM
1.1 . (Fundamenta l Theore m o f Algebra ) The polynomial p(x)
has d roots, counting multiplicities, in the field C of complex numbers.
If th e degre e d i s fou r o r less , the n th e root s ar e function s o f th e coefficient s
which ca n b e expresse d i n term s o f radicals . Her e i s ho w w e ca n produc e thes e
familiar expression s i n the compute r algebr a syste m maple . Reader s mor e familia r
with mathematica, o r reduce , o r other system s wil l find it equall y eas y to perfor m
computations i n thos e compute r algebr a systems .
solve ( a 2 * x~ 2 + a l * x + aO , x ) ;
2 1/2 2 1/2
-al + (a l - 4 a 2 aO ) -a l - (a l - 4 a 2 aO )
1/2 , 1/2
a2 a 2
The followin g expressio n i s one o f the thre e root s o f the genera l cubic :
lprint ( solve ( a 3 * x~ 3 + a 2 * x~ 2 + a l * x + aO , x ) [1] ) ;
!/6/a3*(36*al*a2*a3-108*a0*a3^2-8*a2^3+12*3^(l/2)*(4*al^3*a3
-al^2*a2^2-18*al*a2*a3*a0+27*a0^2*a3^2+4*a0*a2"'3)^(l/2)*a3)
*(1/3)+2/3*(-3*al*a3+a2~2)/a3/(36*al*a2*a3-108*a0*a3~2-8*a2~3
+12*3~ (1/2) *(4*al~3*a3-al~2*a2~2-18*al*a2*a3*a0+27*a(T2*a3~2
+4*a0*a2~3)~(l/2)*a3)~(l/3)-l/3*a2/a3
The polynomia l p(x) ha s d distinc t root s i f an d onl y i f it s discriminant i s
nonzero. Th e discriminan t o f p(x) i s th e produc t o f th e square s o f al l pairwis e
differences o f the roots oip(x). Ca n you spot th e discriminant o f the cubic equatio n
in th e previou s mapl e output ? Th e discriminan t ca n alway s b e expresse d a s a
polynomial i n th e coefficient s ao,Oi,.. . ,a^ . Mor e precisely , i t ca n b e compute d
l
http://dx.doi.org/10.1090/cbms/097/01
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