Multi-homogeneous Bézout theorem

Key takeaways

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

Reference excerpt

In algebra and algebraic geometry, the multi-homogeneous Bézout theorem is a generalization to multi-homogeneous polynomials of Bézout's theorem, which counts the number of isolated common zeros of a set of homogeneous polynomials. This generalization is due to Igor Shafarevich.

Motivation Given a polynomial equation or a system of polynomial equations it is often useful to compute or to bound the number of solutions without computing explicitly the solutions. In the case of a single equation, this problem is solved by the fundamental theorem of algebra, which asserts that the number of complex solutions is bounded by the degree of the polynomial, with equality, if the solutions are counted with their multiplicities. In the case of a system of n polynomial equations in n unknowns, the problem is solved by Bézout's theorem, which asserts that, if the number of complex solutions is finite, their number is bounded by the product of the degrees of the polynomials. Moreover, if the number of solutions at infinity is also finite, then the product of the degrees equals the number of solutions counted with multiplicities and including the solutions at infinity. However, it is rather common that the number of solutions at infinity is infinite. In this case, the product of the degrees of the polynomials may be much larger than the number of roots, and better bounds are useful. Multi-homogeneous Bézout theorem provides such a better bound when the unknowns may be split into several subsets such that the degree of each polynomial in each subset is lower than the total degree of the polynomial. For example, let p 1 , … , p 2 n {\displaystyle p_{1},\ldots ,p_{2n}} be polynomials of degree two which are of degree one in n indeterminate x 1 , … x n , {\displaystyle x_{1},\ldots x_{n},} and also of degree one in y 1 , … y n . {\displaystyle y_{1},\ldots y_{n}.} (that is the polynomials are bilinear. In this case, Bézout's theorem bounds the number of solutions by

2 2 n , {\displaystyle 2^{2n},}

while the multi-homogeneous Bézout theorem gives the bound (using Stirling's approximation)

( 2 n n ) = ( 2 n ) ! ( n ! ) 2 ∼ 2 2 n π n . {\displaystyle {\binom {2n}{n}}={\frac {(2n)!}{(n!)^{2}}}\sim {\frac {2^{2n}}{\sqrt {\pi n}}}.}

Statement A multi-homogeneous polynomial is a polynomial that is homogeneous with respect to several sets of variables. More precisely, consider k positive integers n 1 , … , n k {\displaystyle n_{1},\ldots ,n_{k}} , and, for i = 1, ..., k, the n i + 1 {\displaystyle n_{i}+1} indeterminates x i , 0 , x i , 1 , … , x i , n i . {\displaystyle x_{i,0},x_{i,1},\ldots ,x_{i,n_{i}}.} A polynomial in all these indeterminates is multi-homogeneous of multi-degree d 1 , … , d k , {\displaystyle d_{1},\ldots ,d_{k},} if it is homogeneous of degree d i {\displaystyle d_{i}} in x i , 0 , x i , 1 , … , x i , n i . {\displaystyle x_{i,0},x_{i,1},\ldots ,x_{i,{n_{i}}}.}

A multi-projective variety is a projective subvariety of the product of projective spaces

P n 1 × ⋯ × P n k , {\displaystyle \mathbb {P} _{n_{1}}\times \cdots \times \mathbb {P} _{n_{k}},}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Multi-homogeneous Bézout theorem

Start with the simplest possible case. Write down what Multi-homogeneous Bézout theorem 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 Multi-homogeneous Bézout theorem 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 Multi-homogeneous Bézout theorem 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 Multi-homogeneous Bézout theorem

In research: Multi-homogeneous Bézout theorem 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 Multi-homogeneous Bézout theorem 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: Multi-homogeneous Bézout theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry stubs, Theorems about polynomials, Theorems in algebraic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Multi-homogeneous Bézout theorem 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.

How to study Multi-homogeneous Bézout theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Multi-homogeneous Bézout theorem 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 Multi-homogeneous Bézout theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Multi-homogeneous Bézout theorem in simple terms?

In algebra and algebraic geometry, the multi-homogeneous Bézout theorem is a generalization to multi-homogeneous polynomials of Bézout's theorem, which counts the number of isolated common zeros of a set of homogeneous polynomials. This generalization is due to Igor Shafarevich.

Why does Multi-homogeneous Bézout theorem 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 Multi-homogeneous Bézout theorem?

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 Multi-homogeneous Bézout theorem.