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linearization (Definition)

Linearization is the process of reducing a homogeneous polynomial into a multilinear map over a commutative ring. There are in general two ways of doing this:

  • Method 1. Given any homogeneous polynomial $ f$ of degree $ n$ in $ m$ indeterminates over a commutative scalar ring $ R$ (scalar simply means that the elements of $ R$ commute with the indeterminates).
    Step 1
    If all indeterminates are linear in $ f$, then we are done.
    Step 2
    Otherwise, pick an indeterminate $ x$ such that $ x$ is not linear in $ f$. Without loss of generality, write $ f=f(x,X)$, where $ X$ is the set of indeterminates in $ f$ excluding $ x$. Define $ g(x_1,x_2,X):=f(x_1+x_2,X)-f(x_1,X)-f(x_2,X)$. Then $ g$ is a homogeneous polynomial of degree $ n$ in $ m+1$ indeterminates. However, the highest degree of $ x_1,x_2$ is $ n-1$, one less that of $ x$.
    Step 3
    Repeat the process, starting with Step 1, for the homogeneous polynomial $ g$. Continue until the set $ X$ of indeterminates is enlarged to one $ X^{'}$ such that each $ x\in X^{'}$ is linear.
  • Method 2. This method applies only to homogeneous polynomials that are also homogeneous in each indeterminate, when the other indeterminates are held constant, i.e., $ f(tx,X)=t^nf(x,X)$ for some $ n\in\mathbb{N}$ and any $ t\in R$. Note that if all of the indeterminates in $ f$ commute with each other, then $ f$ is essentially a monomial. So this method is particularly useful when indeterminates are non-commuting. If this is the case, then we use the following algorithm:
    Step 1
    If $ x$ is not linear in $ f$ and that $ f(tx,X)=t^nf(x,X)$, replace $ x$ with a formal linear combination of $ n$ indeterminates over $ R$:
    $\displaystyle r_1x_1+\cdots+r_nx_n$, where $\displaystyle r_i\in R.$
    Step 2
    Define a polynomial $ g\in R\langle x_1,\ldots,x_n \rangle$, the non-commuting free algebra over $ R$ (generated by the non-commuting indeterminates $ x_i$) by:
    $\displaystyle g(x_1,\ldots,x_n):=f(r_1x_1+\cdots+r_nx_n).$
    Step 3
    Expand $ g$ and take the sum of the monomials in $ g$ whose coefficent is $ r_1\cdots r_n$. The result is a linearization of $ f$ for the indeterminate $ x$.
    Step 4
    Take the next non-linear indeterminate and start over (with Step 1). Repeat the process until $ f$ is completely linearized.

Remarks.

  1. The method of linearization is used often in the studies of Lie algebras, Jordan algebras, PI-algebras and quadratic forms.
  2. If the characteristic of scalar ring $ R$ is 0 and $ f$ is a monomial in one indeterminate, we can recover $ f$ back from its linearization by setting all of its indeterminates to a single indeterminate $ x$ and dividing the resulting polynomial by $ n!$:
    $\displaystyle f(x)=\frac{1}{n!}\operatorname{linearization}(f)(x,\ldots,x).$
    Please see the first example below.
  3. If $ f$ is a homogeneous polynomial of degree $ n$, then the linearized $ f$ is a multilinear map in $ n$ indeterminates.

Examples.

  • $ f(x)=x^2$. Then $ f(x_1+x_2)-f(x_1)-f(x_2)=x_1x_2+x_2x_1$ is a linearization of $ x^2$. In general, if $ f(x)=x^n$, then the linearization of $ f$ is
    $\displaystyle \sum_{\sigma\in S_n}x_{\sigma(1)}\cdots x_{\sigma(n)}= \sum_{\sigma\in S_n}\prod_{i=1}^{n}x_{\sigma(i)},$
    where $ S_n$ is the symmetric group on $ \lbrace1,\ldots,n\rbrace$. If in addition all the indeterminates commute with each other and $ n!\neq0$ in the ground ring, then the linearization becomes
    $\displaystyle n!x_1\cdots x_n=\prod_{i=1}^{n}ix_i.$
  • $ f(x,y)=x^3y^2+xyxyx$. Since $ f(tx,y)=t^3f(x,y)$ and $ f(x,ty)=t^2f(x,y)$, $ f$ is homogeneous over $ x$ and $ y$ separately, and thus we can linearize $ f$. First, collect all the monomials having coefficient $ abc$ in $ (ax_1+bx_2+cx_3,y)$, we get
    $\displaystyle g(x_1,x_2,x_3,y):=\sum x_ix_jx_ky^2+x_iyx_jyx_k,$
    where $ i,j,k\in {1,2,3}$ and $ (i-j)(j-k)(k-i)\neq0$. Repeat this for $ y$ and we have
    $\displaystyle h(x_1,x_2,x_3,y_1,y_2):=\sum x_ix_jx_k(y_1y_2+y_2y_1)+(x_iy_1x_jy_2x_k+x_iy_2x_jy_1x_k).$



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Other names:  polarization
Also defines:  linearized

Attachments:
polarization by differential operators (Definition) by rspuzio
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Cross-references: coefficient, ground ring, addition, symmetric group, characteristic, quadratic forms, PI-algebras, Jordan algebras, Lie algebras, sum, expand, generated by, free algebra, polynomial, linear combination, algorithm, monomial, homogeneous, without loss of generality, ring, scalar, commutative, indeterminates, degree, commutative ring, map, multilinear, homogeneous polynomial
There are 11 references to this entry.

This is version 2 of linearization, born on 2004-12-14, modified 2004-12-14.
Object id is 6580, canonical name is Linearization.
Accessed 11930 times total.

Classification:
AMS MSC: 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
 15A69 (Linear and multilinear algebra; matrix theory :: Multilinear algebra, tensor products)
 16R99 (Associative rings and algebras :: Rings with polynomial identity :: Miscellaneous)
 17A99 (Nonassociative rings and algebras :: General nonassociative rings :: Miscellaneous)
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