Symbolic notation Symbolic method

-



1 symbolic notation

1.1 example: discriminant of binary quadratic form
1.2 higher degrees
1.3 more variables





symbolic notation

the symbolic method uses compact rather confusing , mysterious notation invariants, depending on introduction of new symbols a, b, c, ... (from symbolic method gets name) apparently contradictory properties.


example: discriminant of binary quadratic form

these symbols can explained following example (gordan 1887, volume 2, pages 1-3). suppose that








f
(
x
)
=

a

0



x

1


2


+
2

a

1



x

1



x

2


+

a

2



x

2


2





{\displaystyle \displaystyle f(x)=a_{0}x_{1}^{2}+2a_{1}x_{1}x_{2}+a_{2}x_{2}^{2}}



is binary quadratic form invariant given discriminant








Δ
=

a

0



a

2


−

a

1


2


.



{\displaystyle \displaystyle \delta =a_{0}a_{2}-a_{1}^{2}.}



the symbolic representation of discriminant is








2
Δ
=
(
a
b

)

2





{\displaystyle \displaystyle 2\delta =(ab)^{2}}



where , b symbols. meaning of expression (ab) follows. first of all, (ab) shorthand form determinant of matrix rows a1, a2 , b1, b2, so








(
a
b
)
=

a

1



b

2


−

a

2



b

1


.



{\displaystyle \displaystyle (ab)=a_{1}b_{2}-a_{2}b_{1}.}



squaring get








(
a
b

)

2


=

a

1


2



b

2


2


−
2

a

1



a

2



b

1



b

2


+

a

2


2



b

1


2


.



{\displaystyle \displaystyle (ab)^{2}=a_{1}^{2}b_{2}^{2}-2a_{1}a_{2}b_{1}b_{2}+a_{2}^{2}b_{1}^{2}.}



next pretend that








f
(
x
)
=
(

a

1



x

1


+

a

2



x

2



)

2


=
(

b

1



x

1


+

b

2



x

2



)

2





{\displaystyle \displaystyle f(x)=(a_{1}x_{1}+a_{2}x_{2})^{2}=(b_{1}x_{1}+b_{2}x_{2})^{2}}



so that









a

i


=

a

1


2
−
i



a

2


i


=

b

1


2
−
i



b

2


i





{\displaystyle \displaystyle a_{i}=a_{1}^{2-i}a_{2}^{i}=b_{1}^{2-i}b_{2}^{i}}



and ignore fact not seem make sense if f not power of linear form. substituting these values gives








(
a
b

)

2


=

a

2



a

0


−
2

a

1



a

1


+

a

0



a

2


=
2
Δ
.



{\displaystyle \displaystyle (ab)^{2}=a_{2}a_{0}-2a_{1}a_{1}+a_{0}a_{2}=2\delta .}



higher degrees

more if








f
(
x
)
=

a

0



x

1


n


+



(


n
1


)




a

1



x

1


n
−
1



x

2


+
⋯
+

a

n



x

2


n





{\displaystyle \displaystyle f(x)=a_{0}x_{1}^{n}+{\binom {n}{1}}a_{1}x_{1}^{n-1}x_{2}+\cdots +a_{n}x_{2}^{n}}



is binary form of higher degree, 1 introduces new variables a1, a2, b1, b2, c1, c2, properties







f
(
x
)
=
(

a

1



x

1


+

a

2



x

2



)

n


=
(

b

1



x

1


+

b

2



x

2



)

n


=
(

c

1



x

1


+

c

2



x

2



)

n


=
⋯
.


{\displaystyle f(x)=(a_{1}x_{1}+a_{2}x_{2})^{n}=(b_{1}x_{1}+b_{2}x_{2})^{n}=(c_{1}x_{1}+c_{2}x_{2})^{n}=\cdots .}



what means following 2 vector spaces naturally isomorphic:



the vector space of homogeneous polynomials in a0,...an of degree m
the vector space of polynomials in 2m variables a1, a2, b1, b2, c1, c2, ... have degree n in each of m pairs of variables (a1, a2), (b1, b2), (c1, c2), ... , symmetric under permutations of m symbols a, b, ....,

the isomorphism given mapping an−j

1aj

2, bn−j

1bj

2, .... aj. mapping not preserve products of polynomials.


more variables

the extension form f in more 2 variables x1, x2,x3,... similar: 1 introduces symbols a1, a2,a3 , on properties







f
(
x
)
=
(

a

1



x

1


+

a

2



x

2


+

a

3



x

3


+
⋯

)

n


=
(

b

1



x

1


+

b

2



x

2


+

b

3



x

3


+
⋯

)

n


=
(

c

1



x

1


+

c

2



x

2


+

c

3



x

3


+
⋯

)

n


=
⋯
.


{\displaystyle f(x)=(a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}+\cdots )^{n}=(b_{1}x_{1}+b_{2}x_{2}+b_{3}x_{3}+\cdots )^{n}=(c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}+\cdots )^{n}=\cdots .}








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