Preferred notation for systems of d-degree multivariate polynomials
A 2-d matrix is wonderfully practical for manipulating systems of linear equations, multivariate equations of degree 1.
For example, the two equations
$x + y = 4$ and $2x - y = 2$ has matrix notation:
$
left(begin{matrix}
1&1\
2&-1
end{matrix}
right)$
and
$
left(
begin{matrix}
4\
2
end{matrix}
right)$
or
$
left(begin{matrix}
1& 1 & 4\
2&-1 & 2
end{matrix}
right)
$
I am avoiding the concept of matrix multiplication. the notation is simply one row per equation, and one column per variable.
In contrast it is cumbersome to manipulate equations where the degree is higher.
For example operations on $x^3 - x y^2 + 2 x y + 1 = 0$ and $x^2y^2z^2 + x y z - 5 x z^2 - 1 = 0$ can quickly create messy writing.
Is there a preferred schema for writing down such equations that make it easier to manipulate, make the bookkeeping not so onerous, prevent errors? Surely ordering the variables, ordering the degree, and then ordering the monomials is a minimal start. I'm wondering if there is a good way to lay out the monomials.
- How about tuples for monomials where position is
So that $x^3 - x y^2 + 2 x y - 3y + 1 = 0$ is:
$30, -1.12, 2.11, -3.01, 00$
or $x^2y^2z^2 + x y z - 5 x z^2 - 1 = 0$ is:
$222, 111, -5.102, -1.000$
- How about by a matrix-like layout?
Here's an attempt at a layout notation to facilitate multivariate polynomial manipulation.
Let $x^2 - 2x y + y^2 - 1 = 0$ be notated as (limited to degree 2):
$
begin{matrix}\
& 1 & x& x^2 & & w & x w & & w^2 & & \
1 & 1 & .& 1 & & . & . & & . & & \
y & . & -2& & & . & & \
y^2& 1 & & & & & & \
\
z & . & . & & & . & & \
y z& . & &\
\
z^2& . & &\
end{matrix}
$
and $z^2 - 2z w + x z -3 x^2 + w^2 +2 w - 1 = 0$
$
begin{matrix}
& 1 & x& x^2 & & w & x w & & w^2 & & \
1 & 1 & .& -3 & & 2 & . & & 1 & & \
y & . & .& & & . & & \
y^2& . & & & & & & \
\
z & . & 1 & & & -2& & \
y z& . & &\
\
z^2& 1 & &\
end{matrix}
$
This could get cumbersome for larger number of variables, and somewhat overkill for sparse polynomials.
Is there an accepted standard notation, or is simply a one line list of monomials all that is ever used? Are there other notations that ease manipulation of multivariate polynomials by hand?
polynomials notation
edited Dec 5 '18 at 17:26
random123
1,2621720
asked Dec 4 '18 at 16:42
Mitch
6,0212560
add a comment |
A 2-d matrix is wonderfully practical for manipulating systems of linear equations, multivariate equations of degree 1.
For example, the two equations
$x + y = 4$ and $2x - y = 2$ has matrix notation:
$
left(begin{matrix}
1&1\
2&-1
end{matrix}
right)$
and
$
left(
begin{matrix}
4\
2
end{matrix}
right)$
or
$
left(begin{matrix}
1& 1 & 4\
2&-1 & 2
end{matrix}
right)
$
I am avoiding the concept of matrix multiplication. the notation is simply one row per equation, and one column per variable.
In contrast it is cumbersome to manipulate equations where the degree is higher.
For example operations on $x^3 - x y^2 + 2 x y + 1 = 0$ and $x^2y^2z^2 + x y z - 5 x z^2 - 1 = 0$ can quickly create messy writing.
Is there a preferred schema for writing down such equations that make it easier to manipulate, make the bookkeeping not so onerous, prevent errors? Surely ordering the variables, ordering the degree, and then ordering the monomials is a minimal start. I'm wondering if there is a good way to lay out the monomials.
- How about tuples for monomials where position is
So that $x^3 - x y^2 + 2 x y - 3y + 1 = 0$ is:
$30, -1.12, 2.11, -3.01, 00$
or $x^2y^2z^2 + x y z - 5 x z^2 - 1 = 0$ is:
$222, 111, -5.102, -1.000$
- How about by a matrix-like layout?
Here's an attempt at a layout notation to facilitate multivariate polynomial manipulation.
Let $x^2 - 2x y + y^2 - 1 = 0$ be notated as (limited to degree 2):
$
begin{matrix}\
& 1 & x& x^2 & & w & x w & & w^2 & & \
1 & 1 & .& 1 & & . & . & & . & & \
y & . & -2& & & . & & \
y^2& 1 & & & & & & \
\
z & . & . & & & . & & \
y z& . & &\
\
z^2& . & &\
end{matrix}
$
and $z^2 - 2z w + x z -3 x^2 + w^2 +2 w - 1 = 0$
$
begin{matrix}
& 1 & x& x^2 & & w & x w & & w^2 & & \
1 & 1 & .& -3 & & 2 & . & & 1 & & \
y & . & .& & & . & & \
y^2& . & & & & & & \
\
z & . & 1 & & & -2& & \
y z& . & &\
\
z^2& 1 & &\
end{matrix}
$
This could get cumbersome for larger number of variables, and somewhat overkill for sparse polynomials.
Is there an accepted standard notation, or is simply a one line list of monomials all that is ever used? Are there other notations that ease manipulation of multivariate polynomials by hand?
polynomials notation
edited Dec 5 '18 at 17:26
random123
1,2621720
asked Dec 4 '18 at 16:42
Mitch
6,0212560
add a comment |
A 2-d matrix is wonderfully practical for manipulating systems of linear equations, multivariate equations of degree 1.
For example, the two equations
$x + y = 4$ and $2x - y = 2$ has matrix notation:
$
left(begin{matrix}
1&1\
2&-1
end{matrix}
right)$
and
$
left(
begin{matrix}
4\
2
end{matrix}
right)$
or
$
left(begin{matrix}
1& 1 & 4\
2&-1 & 2
end{matrix}
right)
$
I am avoiding the concept of matrix multiplication. the notation is simply one row per equation, and one column per variable.
In contrast it is cumbersome to manipulate equations where the degree is higher.
For example operations on $x^3 - x y^2 + 2 x y + 1 = 0$ and $x^2y^2z^2 + x y z - 5 x z^2 - 1 = 0$ can quickly create messy writing.
Is there a preferred schema for writing down such equations that make it easier to manipulate, make the bookkeeping not so onerous, prevent errors? Surely ordering the variables, ordering the degree, and then ordering the monomials is a minimal start. I'm wondering if there is a good way to lay out the monomials.
- How about tuples for monomials where position is
So that $x^3 - x y^2 + 2 x y - 3y + 1 = 0$ is:
$30, -1.12, 2.11, -3.01, 00$
or $x^2y^2z^2 + x y z - 5 x z^2 - 1 = 0$ is:
$222, 111, -5.102, -1.000$
- How about by a matrix-like layout?
Here's an attempt at a layout notation to facilitate multivariate polynomial manipulation.
Let $x^2 - 2x y + y^2 - 1 = 0$ be notated as (limited to degree 2):
$
begin{matrix}\
& 1 & x& x^2 & & w & x w & & w^2 & & \
1 & 1 & .& 1 & & . & . & & . & & \
y & . & -2& & & . & & \
y^2& 1 & & & & & & \
\
z & . & . & & & . & & \
y z& . & &\
\
z^2& . & &\
end{matrix}
$
and $z^2 - 2z w + x z -3 x^2 + w^2 +2 w - 1 = 0$
$
begin{matrix}
& 1 & x& x^2 & & w & x w & & w^2 & & \
1 & 1 & .& -3 & & 2 & . & & 1 & & \
y & . & .& & & . & & \
y^2& . & & & & & & \
\
z & . & 1 & & & -2& & \
y z& . & &\
\
z^2& 1 & &\
end{matrix}
$
This could get cumbersome for larger number of variables, and somewhat overkill for sparse polynomials.
Is there an accepted standard notation, or is simply a one line list of monomials all that is ever used? Are there other notations that ease manipulation of multivariate polynomials by hand?
polynomials notation
edited Dec 5 '18 at 17:26
random123
1,2621720
asked Dec 4 '18 at 16:42
Mitch
6,0212560
A 2-d matrix is wonderfully practical for manipulating systems of linear equations, multivariate equations of degree 1.
For example, the two equations
$x + y = 4$ and $2x - y = 2$ has matrix notation:
$
left(begin{matrix}
1&1\
2&-1
end{matrix}
right)$
and
$
left(
begin{matrix}
4\
2
end{matrix}
right)$
or
$
left(begin{matrix}
1& 1 & 4\
2&-1 & 2
end{matrix}
right)
$
I am avoiding the concept of matrix multiplication. the notation is simply one row per equation, and one column per variable.
In contrast it is cumbersome to manipulate equations where the degree is higher.
For example operations on $x^3 - x y^2 + 2 x y + 1 = 0$ and $x^2y^2z^2 + x y z - 5 x z^2 - 1 = 0$ can quickly create messy writing.
Is there a preferred schema for writing down such equations that make it easier to manipulate, make the bookkeeping not so onerous, prevent errors? Surely ordering the variables, ordering the degree, and then ordering the monomials is a minimal start. I'm wondering if there is a good way to lay out the monomials.
- How about tuples for monomials where position is
So that $x^3 - x y^2 + 2 x y - 3y + 1 = 0$ is:
$30, -1.12, 2.11, -3.01, 00$
or $x^2y^2z^2 + x y z - 5 x z^2 - 1 = 0$ is:
$222, 111, -5.102, -1.000$
- How about by a matrix-like layout?
Here's an attempt at a layout notation to facilitate multivariate polynomial manipulation.
Let $x^2 - 2x y + y^2 - 1 = 0$ be notated as (limited to degree 2):
$
begin{matrix}\
& 1 & x& x^2 & & w & x w & & w^2 & & \
1 & 1 & .& 1 & & . & . & & . & & \
y & . & -2& & & . & & \
y^2& 1 & & & & & & \
\
z & . & . & & & . & & \
y z& . & &\
\
z^2& . & &\
end{matrix}
$
and $z^2 - 2z w + x z -3 x^2 + w^2 +2 w - 1 = 0$
$
begin{matrix}
& 1 & x& x^2 & & w & x w & & w^2 & & \
1 & 1 & .& -3 & & 2 & . & & 1 & & \
y & . & .& & & . & & \
y^2& . & & & & & & \
\
z & . & 1 & & & -2& & \
y z& . & &\
\
z^2& 1 & &\
end{matrix}
$
This could get cumbersome for larger number of variables, and somewhat overkill for sparse polynomials.
Is there an accepted standard notation, or is simply a one line list of monomials all that is ever used? Are there other notations that ease manipulation of multivariate polynomials by hand?
polynomials notation
polynomials notation
edited Dec 5 '18 at 17:26
random123
1,2621720
asked Dec 4 '18 at 16:42
Mitch
6,0212560
edited Dec 5 '18 at 17:26
random123
1,2621720
asked Dec 4 '18 at 16:42
Mitch
6,0212560
edited Dec 5 '18 at 17:26
random123
1,2621720
edited Dec 5 '18 at 17:26
random123
1,2621720
edited Dec 5 '18 at 17:26
random123
1,2621720
1,2621720
asked Dec 4 '18 at 16:42
Mitch
6,0212560
asked Dec 4 '18 at 16:42
Mitch
6,0212560
asked Dec 4 '18 at 16:42
Mitch
6,0212560
6,0212560
add a comment |
add a comment |
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