Vector and matrix notation is an excellent graphical scheme for doing elementary operations on degree 1 multivariate equations. Adding one equation to another is made easy by ordering the variables and then lining up the coordinates and adding the same coordinates. Dot product is also easy by a similar ordering, multiplying corresponding coords and adding them all up (and matrix multiplication is just an ordered set of dot products.
Slightly differently, one can do arbitrary- order single variable (polynomial) operations by thinking of the polynomial coefficients as the 'digits' (ordered by the degree) and doing operations analogously to binary or decimal calculations (eg long division of polynomials is almost identical to long division of decimal numerals). For example, $(2x-1)(x^2+2x+3)$ is $$ \begin{matrix} & 1& 2 & 3\\ & & 1 & -1&*\\\hline &-1& -2 & -3\\ 1& 2& 3 & 0&+\\ \hline 1& 1& 1 & -3 \end{matrix} $$
Is there a known notation that allows easy manipulation of multinomials of degree larger than 1?
Most presentations of multinomials simply list out the monomial terms (in some standard order like lexicographic) and this suffices for exposition and theory and vague intuition.
But sometimes I struggle with doing actual hand computations on multinomials. For example, solving the equations $x^2 + y^2 - 1 = 0, x + y -1 = 0$ is easy enough, but adding in a extra variable or increasing the degree gets messy.
The combination of the above two techniques might work, but I feel that even though it would be systematic, the schemas might get impractical to manage on paper.
For example, $(x - y - 1)(x+2y+3)$ is $$ \left(\begin{matrix} -1 & 1\\ -1 & \\ \end{matrix} \right) * \left( \begin{matrix} 3 & 1\\ 2 & \\ \end{matrix} \right)\\ = \left( \begin{matrix} -3 & -1\\ -2 & \\ \end{matrix} \right) +\left( \begin{matrix} 0&3 & 1\\ 0&2 & \\ \end{matrix} \right) + \left( \begin{matrix} 0& 0& 0\\ -3 & -1\\ -2 & \\ \end{matrix} \right) = \left( \begin{matrix} -3 & 2& 1\\ -5 & 1& \\ -2\\ \end{matrix} \right) $$
Is that combination indeed a useful notation for elementary operations, or is there another schema that eases manipulation? (or should I not worry about it and just live with lists of monomials)
I'm not looking for algebraic notation with indices and summations, and I'm also not looking for Penrose diagrams which seem to be more abstract. I'm looking for something to ease manipulation of coefficients -and- degree.