Multi-homogeneous polynomial
Okay kiddo, a multi-homogeneous polynomial is a big word that describes a special kind of math expression. Imagine you have a bunch of boxes and each box has different items inside. Each box might have some red apples, green grapes, and juicy oranges, but the number of apples, grapes, and oranges might be different in each box.
Now imagine if we took each box and gave it a special label. The label tells us how many apples, grapes, and oranges are inside that box. For example, one box might be labeled as "2 apples, 3 grapes, and 1 orange".
In math, we use something similar called "degrees". A multi-homogeneous polynomial is an expression that has different parts, and each part has a degree that tells us how many of each variable (like x, y, or z) are in that part.
So, imagine we have an expression like this:
5x^2y^3z + 2xy^4 + 3x^3yz^2
Each part of this expression has different degrees. The first part, 5x^2y^3z, has a degree of 6 (2+3+1=6). The second part, 2xy^4, has a degree of 5 (1+4=5). And the third part, 3x^3yz^2, has a degree of 6 (3+1+2=6).
It's like each part of the expression is a box with a label telling us how many of each variable are inside. And because each part has a degree, we call it "multi-homogeneous" because it has multiple degrees.
So that's it, kiddo! A multi-homogeneous polynomial is just a fancy math expression with different parts that have different degrees, kind of like boxes labeled with how many of each item are inside.
Now imagine if we took each box and gave it a special label. The label tells us how many apples, grapes, and oranges are inside that box. For example, one box might be labeled as "2 apples, 3 grapes, and 1 orange".
In math, we use something similar called "degrees". A multi-homogeneous polynomial is an expression that has different parts, and each part has a degree that tells us how many of each variable (like x, y, or z) are in that part.
So, imagine we have an expression like this:
5x^2y^3z + 2xy^4 + 3x^3yz^2
Each part of this expression has different degrees. The first part, 5x^2y^3z, has a degree of 6 (2+3+1=6). The second part, 2xy^4, has a degree of 5 (1+4=5). And the third part, 3x^3yz^2, has a degree of 6 (3+1+2=6).
It's like each part of the expression is a box with a label telling us how many of each variable are inside. And because each part has a degree, we call it "multi-homogeneous" because it has multiple degrees.
So that's it, kiddo! A multi-homogeneous polynomial is just a fancy math expression with different parts that have different degrees, kind of like boxes labeled with how many of each item are inside.