Multi-index Notation for Differential Operators

Q: Does N include 0 here?

Yes, and it must. The multi-index (0, … ,0) corresponds to the monomial 1 and to the identity operator, both of which are needed. This also matches ISO 80000-2, which defines N to contain 0 .

Q: Is ∂^β(x^α) = 0 the same as |β| > |α|?

No. |β| > |α| is sufficient but not necessary. The exact criterion is that β fails to be componentwise below α ; for instance ∂_2(x_1^5) = 0 although |β| = 1 and |α| = 5 .

Q: How do I remember formula (2.3)?

Read it as a Leibniz rule. Moving ∂^β past x^α , each derivative either passes through untouched or lands on the monomial. The index γ records which derivatives land: βγ counts the ways of choosing them, and α!/(α-γ)! is what they produce.

Q: Why do the counting formulas use 2n?

Because a canonical basis element of A_n is indexed by a pair of multi-indices in n variables, that is by one multi-index in 2n variables. The degree is its length, so counting operators of degree at most m is counting lattice points of length at most m in N^2n , giving m+2n2n .

Overview

Working in the Weyl algebra $A_n$ means writing products such as $x_1^3x_2^0x_3^5,∂_1^2∂_2^4∂_3^0$ over and over. Written out, these expressions are unreadable and the patterns in them are invisible. Multi-index notation replaces each block of exponents by a single vector: $α = (3,0,5)$ , $β = (2,4,0)$ , and the operator becomes $x^α∂^β$ .

This is more than an abbreviation. Once exponents are vectors, statements about differential operators become statements about $N^n$ , and the combinatorics can be done in the exponent lattice. The canonical basis of $A_n$ is indexed by $N^n × N^n = N^2n$ ; the degree of an operator is a maximum of lengths of multi-indices; counting basis elements becomes counting lattice points in a simplex; and the general Leibniz rule becomes a sum over the componentwise interval below a multi-index.

Two facts about the notation deserve emphasis at the outset, because both are common sources of confusion. First, multi-indices carry two different orderings - the partial order by components and the total preorder by length - and almost every statement in the subject depends on which is meant. Second, $x^α∂^β$ denotes a composite operator, not a function: it means multiply by $x^α$ after differentiating, whereas $∂^β(x^α)$ means differentiate the polynomial $x^α$ . Those two expressions are different objects and are almost never equal.

This page collects the definitions, the identities that get used constantly, and the counting formulas that later dimension arguments rely on.

Definition

Multi-index and its invariants

A multi-index in $n$ variables is an element $α = (α_1, … ,α_n)$ of $N^n$ , where $N = 0,1,2, …$ . Attached to it are

its length $|α| = α_1 + α_2 + … + α_n$ ;
its factorial $α! = α_1!,α_2! ⋅ sα_n!$ ;
the monomial $x^α = x_1^α_1x_2^α_2 ⋅ s x_n^α_n ∈ K[X]$ ;
the iterated derivative $∂^α = ∂_1^α_1∂_2^α_2 ⋅ s ∂_n^α_n$ .

Because the $x_i$ commute with each other and the $∂_i$ commute with each other, neither $x^α$ nor $∂^α$ depends on the order in which the factors are written.

Order, sum and binomial coefficient

For multi-indices $α, γ ∈ N^n$ write $γ ≤ α$ when $γ_i ≤ α_i$ for every $i$ . Sums and differences are componentwise, with $α - γ$ defined only when $γ ≤ α$ . The multi-index binomial coefficient is

αγ = ∏ _i=1^n α_iγ_i = (α!)/(γ!,(α-γ)!) (γ ≤ α).

Finally, $e_i$ denotes the multi-index with $1$ in position $i$ and $0$ elsewhere, so that $x^e_i = x_i$ and $∂^e_i = ∂_i$ .

Note

A pair $(α,β)$ of multi-indices in $N^n$ is itself a multi-index in $N^2n$ , of length $|α| + |β|$ . That observation is what lets the degree of an element of $A_n$ be defined by a single length, and it is why the counting formulas below are stated in $2n$ variables when they are applied to $A_n$ .

Core Concepts

The exponent lattice

Think of a monomial not as a product but as the point of $N^n$ recording its exponents. Multiplying monomials adds points; the total degree is the $ℓ^1$ length; divisibility of one monomial by another is the componentwise order. Under this dictionary, questions about polynomials become questions about lattice points, and a surprising number of them become elementary counting problems.

Two orders, doing different jobs

Ordering by length is total but coarse: $(2,0)$ and $(0,2)$ and $(1,1)$ all have length $2$ . Ordering componentwise is fine but partial: $(2,0)$ and $(0,2)$ are incomparable. Each order governs different statements. Derivatives are controlled by the componentwise order - $∂^β(x^α)$ is non-zero exactly when $β ≤ α$ - while filtrations and degrees are controlled by length. Confusing them produces false statements that look plausible.

Operators versus values

In $A_n$ the symbol $x^α$ means the operator 'multiply by $x^α$ ', so $x^α ∂^β$ is a composition of operators. Applying that composition to a polynomial $f$ gives $x^α ⋅ ∂^β(f)$ . The parenthesis is doing real work: $∂^β x^α$ without parentheses is the operator obtained by composing in the other order, and by the commutation relations it equals $x^α∂^β$ plus strictly lower-degree corrections.

Why this notation and not tensor indices

Multi-index notation is adapted to commuting symbols. It compresses $x_1^α_1 ⋅ s x_n^α_n$ into one symbol precisely because reordering the factors changes nothing. That is why it is used for the variables and for the derivatives separately, but never across the two blocks: $x$ 's and $∂$ 's do not commute, so no single multi-index in $N^2n$ can stand for an arbitrary word in the generators. The pair $(α,β)$ stands for the canonical arrangement only.

Key Equations

The identity used most often is the value of an iterated derivative on a monomial. For $α, β ∈ N^n$ ,

∂^β (x^α) = cases α!(α-β)!; x^α-β & if β ≤ α, [4pt] 0 & otherwise. cases

(2.1)

The special case $β = α$ gives $∂^α(x^α) = α!$ , the fact that drives the proof that the canonical basis is a basis. The general Leibniz rule reads

∂^α (fg) = ∑ _γ ≤ α αγ, ∂^γ(f), ∂^α-γ(g), f,g ∈ K[X].

(2.2)

Its operator form, obtained by taking $f = x^α$ and reading both sides as elements of $A_n$ , is the rule for straightening a product:

∂^β, x^α = ∑ _γ ≤ (α,β) βγ, (α!)/((α-γ)!); x^α-γ, ∂^β-γ.

(2.3)

The multinomial theorem and the Taylor expansion of a polynomial both become one-line statements:

(x_1 + … + x_n)^m = ∑ _|α| = m (m!)/(α!), x^α, f(x) = ∑ _α (∂^α f(a))/(α!),(x-a)^α.

(2.4)

Finally the two counting formulas. The number of multi-indices of exact length $m$ and of length at most $m$ in $n$ variables are

#α ∈ N^n : |α| = m = m+n-1n-1, #α ∈ N^n : |α| ≤ m = m+nn.

(2.5)

Applied with $2n$ in place of $n$ , the second formula counts the canonical basis elements of $A_n$ of degree at most $m$ , giving $m+2n2n$ - the starting point of every dimension computation later in the theory.

Variable Definitions

$n$: the number of variables
$α, β, γ, σ$: multi-indices, elements of $N^n$
$α_i$: the $i$ -th entry of the multi-index $α$ , a non-negative integer
$|α|$: the length $α_1 + … + α_n$
$α!$: the factorial $α_1! ⋅ sα_n!$
$γ ≤ α$: the componentwise order: $γ_i ≤ α_i$ for every $i$
$(\binom{α}{γ})$: the product of the entrywise binomial coefficients
$e_i$: the multi-index with $1$ in position $i$ , zeros elsewhere
$x^α$: the monomial $x_1^α_1 ⋅ s x_n^α_n$ , or the operator multiplying by it
$∂^β$: the iterated partial derivative $∂_1^β_1 ⋅ s∂_n^β_n$

Properties and Behaviour

Basic identities

$x^α x^β = x^α+β$ and $∂^α ∂^β = ∂^α+β$ ; the exponents add.
$|α + β| = |α| + |β|$ , so length is additive.
$_x(x^α) = |α|$ , and the monomials $x^α : |α| ≤ m$ are a basis of the polynomials of degree at most $m$ .
$(α+β)! ≠ α!,β!$ in general; the correct relation is $α+βα = (α+β)!/(α!,β!)$ , an integer.
$∂^β(x^α) = 0$ whenever $|β| > |α|$ , but the converse fails: $∂^(1,0)(x_2^5) = 0$ with $|β| = 1 < 5$ .

Lengths bound components

If $γ ≤ α$ then $|γ| ≤ |α|$ , but not conversely. Consequently every statement proved using the componentwise order implies the corresponding statement about lengths, and no statement about lengths implies anything componentwise. The one useful partial converse: if $γ ≤ α$ and $|γ| = |α|$ then $γ = α$ . This is the pigeonhole step at the heart of the canonical basis proof.

Degree in the Weyl algebra

For $D = ∑ c_αβx^α∂^β ≠ 0$ in canonical form, $D = |α| + |β| : c_αβ ≠ 0$ , with $0 = - ∈ fty$ . Straightening formula (2.3) shows the correction terms all have $|α - γ| + |β - γ| = |α| + |β| - 2|γ|$ , strictly smaller for $γ ≠ 0$ : reordering an operator lowers degree by an even amount, never raises it. This is the multi-index reason behind the commutator bound $[D,D'] ≤ D + D' - 2$ .

Examples and Special Cases

Reading a multi-index

With $n = 3$ and $α = (2,0,1)$ : $|α| = 3$ , $α! = 2! ⋅ 0! ⋅ 1! = 2$ , $x^α = x_1^2 x_3$ , and $∂^α = ∂_1^2∂_3$ . Note $x_2$ does not appear, because its exponent is zero; a multi-index always has $n$ entries even when most are zero.

Formula (2.1) in action

Take $n = 2$ , $α = (3,2)$ , $β = (1,2)$ . Since $β ≤ α$ , formula (2.1) gives $α!/(α-β)! = (3! ⋅ 2!)/(2! ⋅ 0!) = 12/2 = 6$ and $∂_1∂_2^2(x_1^3x_2^2) = 6, x_1^2$ . Checking directly: $∂_2^2(x_1^3x_2^2) = 2x_1^3$ , then $∂_1(2x_1^3) = 6x_1^2$ .

Incomparable indices

$α = (2,0)$ and $β = (0,2)$ have the same length but neither is componentwise below the other. Accordingly $∂^β(x^α) = ∂_2^2(x_1^2) = 0$ and $∂^α(x^β) = 0$ as well, while $∂^α(x^α) = 2$ .

Counting

In two variables there are $2+11 = 3$ multi-indices of length exactly $2$ , namely $(2,0), (1,1), (0,2)$ , and $2+22 = 6$ of length at most $2$ , adding $(0,0), (1,0), (0,1)$ . Both match formula (2.5).

Worked Example

Straightening $∂_1∂_2, x_1^2 x_2$ in $A_2$ , and counting a filtration piece

Step 1 - name the indices
Take $n = 2$ , $α = (2,1)$ and $β = (1,1)$ , so that $x^α = x_1^2x_2$ and $∂^β = ∂_1∂_2$ . We want the canonical form of the operator $∂^β x^α$ . Here $(α,β) = (1,1)$ , so $γ$ in (2.3) ranges over $(0,0), (1,0), (0,1), (1,1)$ .
Step 2 - evaluate the four terms
Every binomial coefficient $(\binom{β}{γ})$ equals $1$ here, since each entry of $β$ is $1$ . With $α! = 2! ⋅ 1! = 2$ , the factor $α!/(α-γ)!$ is:
- $γ = (0,0)$ : $(α-γ)! = 2$ , factor $1$ , term $x_1^2x_2,∂_1∂_2$ ;
- $γ = (1,0)$ : $(α-γ)! = 1! ⋅ 1! = 1$ , factor $2$ , term $2,x_1x_2,∂_2$ ;
- $γ = (0,1)$ : $(α-γ)! = 2! ⋅ 0! = 2$ , factor $1$ , term $x_1^2,∂_1$ ;
- $γ = (1,1)$ : $(α-γ)! = 1! ⋅ 0! = 1$ , factor $2$ , term $2,x_1$ .
$∂_1∂_2, x_1^2x_2 = x_1^2x_2,∂_1∂_2 + 2,x_1x_2,∂_2 + x_1^2,∂_1 + 2,x_1.$
Step 3 - verify on test polynomials
Apply both sides to $1$ : the left side is $∂_1∂_2(x_1^2x_2) = ∂_1(x_1^2) = 2x_1$ ; on the right only the last term survives, giving $2x_1$ .

Apply both sides to $x_2$ : the left side is $∂_1∂_2(x_1^2x_2^2) = ∂_1(2x_1^2x_2) = 4x_1x_2$ ; the right side gives $0 + 2x_1x_2 + 0 + 2x_1x_2 = 4x_1x_2$ .

Apply both sides to $x_1x_2$ : the left side is $∂_1∂_2(x_1^3x_2^2) = ∂_1(2x_1^3x_2) = 6x_1^2x_2$ ; the right side gives $x_1^2x_2 + 2x_1^2x_2 + x_1^2x_2 + 2x_1^2x_2 = 6x_1^2x_2$ . Three independent checks agree.
Step 4 - count the operators of degree at most 3
The operator just computed has degree $|α| + |β| = 3$ . By (2.5) with $2n = 4$ , the space of elements of $A_2$ of degree at most $3$ has dimension $3+44 = 74 = 35$ .

Cross-check by summing over exact degrees, using the first formula in (2.5) with $n$ replaced by $4$ : degrees $0,1,2,3$ contribute $33 = 1$ , $43 = 4$ , $53 = 10$ and $63 = 20$ . The total is $1+4+10+20 = 35$ .

Result

$∂_1∂_2,x_1^2x_2 = x_1^2x_2,∂_1∂_2 + 2x_1x_2,∂_2 + x_1^2,∂_1 + 2x_1$ , an element of degree $3$ in a space of dimension $35$ . Every correction term has degree $3 - 2|γ|$ , dropping by two at each step, exactly as the degree bound on commutators predicts.

Applications and Industry Use

In a mathematics topic, this section covers downstream use inside mathematics, computing and engineering rather than a manufactured product.

Indexing the canonical basis. The basis of $A_n$ is $x^α∂^β$ indexed by $N^2n$ , so an operator is a finitely supported function $N^2n → K$ . Every algorithm in the subject manipulates that function.
Filtrations. The Bernstein filtration collects the $x^α∂^β$ with $|α| + |β| ≤ m$ ; the order filtration collects those with $|β| ≤ m$ . Both are conditions on multi-index lengths, which is why their Hilbert functions are lattice point counts.
Partial differential equations. A linear PDE with polynomial coefficients is written $∑ _|β| ≤ k a_β(x)∂^β u = 0$ ; the notation is standard in analysis for the same reasons it is standard here.
Symbols and characteristic varieties. Replacing $∂^β$ by $ξ^β$ in the top-degree part turns an operator into a polynomial on the cotangent space; the substitution is literally a relabelling of multi-indices.
Computer algebra. Sparse polynomial and operator arithmetic is implemented as arithmetic on exponent vectors with hashed lookup, and term orders are orders on $N^2n$ .

Standards and Codes

For mathematics, the relevant standards are notation, numeric and markup standards together with reference implementations.

ISO 80000-2 fixes $ℕ$ for the non-negative integers - note that the standard includes $0$ , which is what makes $N^n$ the right index set here - and $∂$ for partial differentiation.
ISO/IEC 40314 (MathML 3.0) is the encoding used for the expressions on this page, so that a superscripted multi-index remains machine-readable rather than becoming an image.
There is no standard for multi-index notation itself; the conventions used here (length written $|α|$ , factorial written $α!$ , componentwise order written $≤$ ) are the ones common to the analysis and D-module literature and are the ones Coutinho uses.
One genuine notational divergence to watch: some texts write $D^α$ or $∂_x^α$ for $∂^α$ , and some analysts insert a factor of $i$ or $2π i$ into $D^α$ to make Fourier transforms cleaner. Those conventions are not used in this collection.

Computational Notes

Read this as the manufacturing section of the template: how the object is actually built by machine, at what cost, and where the computation stops being decidable.

An operator in $A_n$ is stored as a dictionary keyed by the pair $(α,β) ∈ N^2n$ , with $K$ -coefficients as values. Addition merges dictionaries; multiplication applies (2.3) term by term. Three practical points follow from the notation itself.

Multiplying $x^α∂^β$ by $x^α'∂^β'$ produces one term for each $γ ≤ (α',β)$ , so the number of output terms is $∏ _i ( (α'_i,β_i)+1)$ . Sparse operators of low degree multiply cheaply; dense high-degree ones do not.
The leading exponent pair of a product is the sum of the leading exponent pairs, because all correction terms have strictly smaller length. This is what makes term orders on $N^2n$ compatible with multiplication and non-commutative Groebner bases possible.
Factorial ratios $α!/(α-γ)!$ should be computed as falling factorials $∏ _i α_i(α_i-1) ⋅ s(α_i-γ_i+1)$ rather than as quotients of factorials, to avoid large intermediate integers.

Macaulay2's Dmodules package, Singular's Plural/dmod.lib and SageMath's ore_algebra all use exponent-vector representations of this kind, and their documentation states the same convention: variables left, derivatives right.

Failure Modes and Common Mistakes

Confusing $α!$ with $|α|!$

For $α = (2,1)$ , $α! = 2$ but $|α|! = 3! = 6$ . Both appear in the subject - $α!$ in (2.1) and (2.2), $|α|!$ in the multinomial coefficient $m!/α!$ of (2.4) - and substituting one for the other silently corrupts every subsequent constant.

Assuming $|β| ≤ |α|$ implies $β ≤ α$

It does not. With $β = (0,1)$ and $α = (2,0)$ we have $|β| = 1 < 2 = |α|$ , yet $∂^β(x^α) = ∂_2(x_1^2) = 0$ . Formula (2.1) is governed by the componentwise order; only the vanishing statement $|β| > |α| ⇒ ∂^β(x^α) = 0$ is governed by length.

Reading $x^α∂^β$ as a function value

$x^α∂^β$ is an operator; $∂^β(x^α)$ is a polynomial. In $A_1$ with $α = β = 1$ , the operator $x∂$ sends $f$ to $x f^{'}$ , whereas $∂(x) = 1$ . Writing the two the same way, or cancelling the $x^α$ against the $∂^β$ , is the single most common slip when first working in $A_n$ .

Dropping zero entries

A multi-index in $n$ variables always has $n$ entries. Writing $α = (2,1)$ when $n = 3$ is undefined; the intended index is $(2,1,0)$ . This matters when multi-indices are compared or added across expressions in different numbers of variables - for instance when $A_m$ is viewed inside $A_n$ - where the shorter index must be padded with zeros.

Subtracting outside the allowed range

$α - γ$ is a multi-index only when $γ ≤ α$ ; otherwise it has a negative entry and $x^α-γ$ is not a polynomial. Sums such as (2.2) and (2.3) restrict the summation index for exactly this reason, and an implementation that omits the range check will silently produce Laurent monomials.

Historical Notes

Compressed exponent notation appears wherever many variables are differentiated many times, and it became standard through partial differential equations rather than algebra: Schwartz's theory of distributions in the late 1940s and Hormander's treatments of linear PDE fixed the now-universal conventions $x^α$ , $∂^α$ , $|α|$ and $α!$ . Algebraists inherited the notation together with the subject matter.

For the Weyl algebra the notation does something extra. Because the canonical basis is indexed by $N^2n$ , questions about $A_n$ can be transported to the commutative polynomial ring in $2n$ variables, whose combinatorics is classical. That transport - degrees to lengths, filtration pieces to simplices of lattice points, operators to their symbols - is the technical device behind the dimension theory of Chapters 7 to 9 of Coutinho's book, and it is available only because the notation makes the index set visible.

Comparison

The two orderings of multi-indices and what each controls.
	Componentwise order $γ ≤ α$	Order by length $\|γ\| ≤ \|α\|$
Type of order	Partial	Total on lengths, but many indices share a length
Controls	When $∂^γ(x^α) ≠ 0$ ; divisibility of monomials	Degree, filtrations, Hilbert functions
Typical statement	$∂^γ(x^α) ≠ 0 ⟺ γ ≤ α$	$(DD') = D + D'$
Incomparable pairs	$(2,0)$ and $(0,2)$	None; lengths are integers
Used in	Leibniz rule, straightening formula (2.3)	Bernstein filtration, degree, dimension theory
Relation	$γ ≤ α$ implies $\|γ\| ≤ \|α\|$	The converse is false

Key Takeaways

Key points

A multi-index is a vector $α ∈ N^n$ ; it carries a length $|α|$ , a factorial $α!$ , a monomial $x^α$ and a derivative $∂^α$ .
The componentwise order decides when $∂^β(x^α)$ is non-zero; the length decides degrees and filtrations. They are different orders and are not interchangeable.
$∂^β(x^α) = α!/(α-β)!, x^α-β$ when $β ≤ α$ , and $0$ otherwise; in particular $∂^α(x^α) = α!$ .
Straightening in $A_n$ is the single formula (2.3), whose correction terms drop degree by $2|γ|$ .
A pair $(α,β)$ is a multi-index in $N^2n$ , so counting elements of $A_n$ of degree at most $m$ gives $m+2n2n$ .
$x^α∂^β$ is an operator and $∂^β(x^α)$ is a polynomial; keeping them apart avoids most beginners' errors.

FAQs

Does

ℕ

include

0

here?

Yes, and it must. The multi-index $(0, … ,0)$ corresponds to the monomial $1$ and to the identity operator, both of which are needed. This also matches ISO 80000-2, which defines $ℕ$ to contain $0$ .

Why is there no single multi-index for a general element of

A_n

Because the generators do not all commute. A multi-index compresses a product of commuting factors; the $x$ 's commute among themselves and the $∂$ 's do too, so each block gets its own index, but a word mixing them is not determined by its exponent counts. The pair $(α,β)$ specifies a canonically ordered product only.

∂^β(x^α) = 0

the same as

|β| > |α|

No. $|β| > |α|$ is sufficient but not necessary. The exact criterion is that $β$ fails to be componentwise below $α$ ; for instance $∂_2(x_1^5) = 0$ although $|β| = 1$ and $|α| = 5$ .

How do I remember formula (2.3)?

Read it as a Leibniz rule. Moving $∂^β$ past $x^α$ , each derivative either passes through untouched or lands on the monomial. The index $γ$ records which derivatives land: $(\binom{β}{γ})$ counts the ways of choosing them, and $α!/(α-γ)!$ is what they produce.

Why do the counting formulas use

2n

Because a canonical basis element of $A_n$ is indexed by a pair of multi-indices in $n$ variables, that is by one multi-index in $2n$ variables. The degree is its length, so counting operators of degree at most $m$ is counting lattice points of length at most $m$ in $N^2n$ , giving $m+2n2n$ .

Does any of this depend on the characteristic of

K

The notation does not, and formulas (2.1) to (2.3) hold over any commutative ring, with the integers $α!$ and $(\binom{α}{γ})$ read as multiples of $1$ . What changes in characteristic $p$ is that these integers can become zero in $K$ - for example $∂^(p)(x^(p)) = p! = 0$ - which is why the canonical basis result needs characteristic zero even though the notation does not.

What is the relationship to the notation used in PDE textbooks?

It is the same notation. A linear operator with polynomial coefficients written $∑ _|β| ≤ k a_β(x)∂^β$ in an analysis text is exactly an element of $A_n$ in canonical form, grouped by $β$ . The difference is that analysis usually allows smooth or holomorphic coefficients, whereas $A_n$ insists on polynomial ones.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 1 §2, where multi-index notation is introduced and used for the canonical basis (1.2.1) and Lemma (1.2.2).
L. Hormander, The Analysis of Linear Partial Differential Operators I, 2nd edition, Springer, 1990 - §1.1, for the standard analytic conventions.
L. Schwartz, Theorie des distributions, Hermann, Paris, 1950-1951 - the source of much of the modern notation for iterated partial derivatives.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics 30, revised edition, American Mathematical Society, 2001 - Ch. 1, for the Weyl algebra in this notation.
M. Saito, B. Sturmfels and N. Takayama, Groebner Deformations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics 6, Springer, 2000 - Ch. 1, for exponent-vector representations and term orders on $N^2n$ .
R. P. Stanley, Enumerative Combinatorics, Volume 1, 2nd edition, Cambridge University Press, 2011 - Ch. 1, for the lattice point counts (2.5).
ISO 80000-2:2019, Quantities and units - Part 2: Mathematics, International Organization for Standardization.
ISO/IEC 40314:2016, Information technology - Mathematical Markup Language (MathML) Version 3.0, International Organization for Standardization.

AI Suggested Questions

Verify formula (2.3) for $n = 1$ , $α = 3$ , $β = 2$ , and check the result by applying both sides to $1$ , $x$ and $x^2$ .
Count the elements of the canonical basis of $A_3$ of degree exactly $4$ , and check the answer against the total for degree at most $4$ .
Prove the general Leibniz rule (2.2) by induction on $|α|$ .
Show that $α+βα$ is an integer for all multi-indices, and interpret it as a count.
Explain why the number of terms produced by one application of (2.3) is $∏ _i( (α_i,β_i)+1)$ .
Give an example of multi-indices with $|γ| ≤ |α|$ , $γ ≠ α$ , and $∂^γ(x^α) ≠ 0$ , and one with $∂^γ(x^α) = 0$ .
Describe a term order on $N^2n$ that refines total degree, and check that it is compatible with multiplication in $A_n$ .