# Standard form polynomial

The calculator converts a multivariate polynomial to the standard form.

The calculator further presents a multivariate polynomial in the standard form (expands parentheses, exponentiates, and combines similar terms). Polynomial variables can be specified in lowercase English letters or using the exponent tuple form. For example, the following two notations equal: 3a^2bd + c and 3 [2 1 0 1] + [0 0 1]. You can choose output variables representation to the symbolic form, indexed variables form, or the tuple of exponents. The calculator also gives the degree of the polynomial and the vector of degrees of monomials. The coefficients of the resulting polynomial can be calculated in the field of rational or real numbers.

#### Standard Form Polynomial

Result

Polynomial degree

Monomial degrees

### Monomial

A monomial is is a product of powers of several variables xi with nonnegative integer exponents ai:
$x^{\alpha}={x_1}^{\alpha_1}{x_2}^{\alpha_2}{x_3}^{\alpha_3} ... {x_n}^{\alpha_n}$
If the number of variables is small, polynomial variables can be written by latin letters. E.g. x12x2 and x2y are - equivalent notation of the two-variable monomial.
A monomial can also be represented as a tuple of exponents:
$\alpha=({\alpha_1},{\alpha_2},{\alpha_3}, ... ,{\alpha_n})$
E.g. x2y3z monomial can be represented as tuple: (2,3,1)
The monomial degree is the sum of all variable exponents:
$\mid \alpha \mid = \alpha_1 + \alpha_2 + \alpha_3 + ... + \alpha_n$
E.g., degree of monomial: x2y3z is 2+3+1 = 6

### Polynomial

A polynomial is a finite sum of monomials multiplied by coefficients cI:
$f=\sum _I c_I {x_1}^{\alpha_1}{x_2}^{\alpha_2}{x_3}^{\alpha_3} ... {x_n}^{\alpha_n}$
A polynomial degree deg(f) is the maximum of monomial degree |α| with nonzero coefficients.
Unlike polynomials of one variable, multivariate polynomials can have several monomials with the same degree.
In this regard, the question arises of determining the order on the set of terms of the polynomial.

### Monomial order1

There are several ways to specify the order of monomials.

#### Lexicographic order

The simplest monomial order is lexicographic. In this case, the leftmost nonzero coordinate of the vector obtained by subtracting the exponent tuples of the compared monomials is positive:
$x^{\alpha}>_{lex}x^{\beta} \Leftarrow {\alpha}>{\beta}$
Lexicographic order example:
$x^{\alpha}=x^2y^3z >_{lex} x^{\beta}=x^2y^2z^3, \\\alpha-\beta=(2,3,1)-(2,2,3)=(0,1,-2)$
The first monomial xα is lexicographically greater than second one xβ, since after subtraction of exponent tuples we obtain (0,1,-2), where leftmost nonzero coordinate is positive.

The graded lexicographic order is determined primarily by the degree of the monomial. If the degree is greater, then the monomial is also considered greater. In the case of equal degrees, lexicographic comparison is applied:
$x^{\alpha}>_{grlex}x^{\beta} \Leftarrow \begin{cases} \mid{\alpha}\mid>\mid{\beta}\mid \\ \mid{\alpha}\mid=\mid{\beta}\mid, {\alpha}>_{lex} {\beta} \end{cases}$
a)
$x^{\beta}=x^2y^2z^3 >_{grlex} x^{\alpha}=x^2y^3z , \\ \mid\beta\mid = 7 > \mid\alpha\mid=6$
The monomial xβ is greater than xα, since the degree |β|=7 is greater than the degree |α|=6.
b)
$x^{\alpha}=x^2y^3z >_{grlex} x^{\gamma}=xy^5 , \\ \mid\alpha\mid = \mid\gamma\mid=6, {\alpha}>{\gamma}$
The monomial xα is greater than the xγ, since they are of the same degree, but the first is greater than the second lexicographically.

The graded reverse lexicographic order is similar to the previous one. If the degree is greater, then the monomial is also considered greater. The monomial is greater if the rightmost nonzero coordinate of the vector obtained by subtracting the exponent tuples of the compared monomials is negative in the case of equal degrees.
Examples of graded reverse lexicographic comparison:
a)
$x^{\beta}=x^2y^2z^3 >_{grevlex} x^{\alpha}=x^2y^3z , \\ \mid\beta\mid = 7 > \mid\alpha\mid=6$
The monomial xβ is greater than xα, since degree |β|=7 is greater than degree |α|=6.
b)
$x^{\gamma}=xy^5 >_{grevlex} x^{\alpha}=x^2y^3z , \\ \mid\alpha\mid = \mid\gamma\mid=6, {\gamma}-{\alpha}=(1,5,0)-(2,3,1)=(-1,2,-1)$
The monomial xγ is greater than the xα, since their degrees are equal, but the subtraction of exponent tuples gives (-1,2,-1) and we see the rightmost value is below the zero.

1. David Cox, John Little, Donal O’Shea Ideals, Varieties, and
Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Third Edition, 2007, Springer

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