mathematics

Hilbert–Samuel function

Hilbert–Samuel function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hilbert–Samuel function rather than just read about it. In short: In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module M {\displaystyle M} over a commutative Noetherian local ring A {\displaystyle A} and a primary ideal I {\displaystyle I} of A {\displaystyle A} is the map χ M I : N → N {\displaystyle \chi _{M}^{I}:\mathbb {N} \rightarrow \mathbb {N} } such that, for all n ∈ N {\displaystyle n\in \m…

Key takeaways

Hilbert–Samuel function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hilbert–Samuel function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hilbert–Samuel function from memory before moving on to harder problems.

Reference excerpt

In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module M {\displaystyle M} over a commutative Noetherian local ring A {\displaystyle A} and a primary ideal I {\displaystyle I} of A {\displaystyle A} is the map χ M I : N → N {\displaystyle \chi _{M}^{I}:\mathbb {N} \rightarrow \mathbb {N} } such that, for all n ∈ N {\displaystyle n\in \mathbb {N} } ,

χ M I ( n ) = ℓ ( M / I n M ) {\displaystyle \chi _{M}^{I}(n)=\ell (M/I^{n}M)}

where ℓ {\displaystyle \ell } denotes the length over A {\displaystyle A} . It is related to the Hilbert function of the associated graded module gr I ⁡ ( M ) {\displaystyle \operatorname {gr} _{I}(M)} by the identity

χ M I ( n ) = ∑ i = 0 n H ( gr I ⁡ ( M ) , i ) . {\displaystyle \chi _{M}^{I}(n)=\sum _{i=0}^{n}H(\operatorname {gr} _{I}(M),i).}

For sufficiently large n {\displaystyle n} , it coincides with a polynomial function of degree equal to dim ⁡ ( gr I ⁡ ( M ) ) {\displaystyle \dim(\operatorname {gr} _{I}(M))} , often called the Hilbert-Samuel polynomial (or Hilbert polynomial).

Examples For the ring of formal power series in two variables k [ [ x , y ] ] {\displaystyle k[[x,y]]} taken as a module over itself and the ideal I {\displaystyle I} generated by the monomials x2 and y3 we have

χ ( 1 ) = 6 , χ ( 2 ) = 18 , χ ( 3 ) = 36 , χ ( 4 ) = 60 , and in general χ ( n ) = 3 n ( n + 1 ) for n ≥ 0. {\displaystyle \chi (1)=6,\quad \chi (2)=18,\quad \chi (3)=36,\quad \chi (4)=60,{\text{ and in general }}\chi (n)=3n(n+1){\text{ for }}n\geq 0.}

Degree bounds Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence. However, it is still reasonably close to being additive, as a consequence of the Artin–Rees lemma. We denote by P I , M {\displaystyle P_{I,M}} the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.

Proof: Tensoring the given exact sequence with R / I n {\displaystyle R/I^{n}} and computing the kernel we get the exact sequence:

0 → ( I n M ∩ M ′ ) / I n M ′ → M ′ / I n M ′ → M / I n M → M ″ / I n M ″ → 0 , {\displaystyle 0\to (I^{n}M\cap M')/I^{n}M'\to M'/I^{n}M'\to M/I^{n}M\to M''/I^{n}M''\to 0,}

which gives us:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hilbert–Samuel function

Start with the simplest possible case. Write down what Hilbert–Samuel function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hilbert–Samuel function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hilbert–Samuel function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hilbert–Samuel function

In research: Hilbert–Samuel function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hilbert–Samuel function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hilbert–Samuel function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, Commutative algebra, so understanding it makes those chapters shorter.
In everyday life: Look for Hilbert–Samuel function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hilbert–Samuel function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hilbert–Samuel function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hilbert–Samuel function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hilbert–Samuel function in simple terms?

Why does Hilbert–Samuel function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hilbert–Samuel function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hilbert–Samuel function.