Sparse polynomial

Key takeaways

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

Reference excerpt

In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables would suggest. For example, x 10 + 3 x 3 + 1 {\displaystyle x^{10}+3x^{3}+1} is a sparse polynomial, as it is a trinomial with a degree of 10 {\displaystyle 10} . The motivation for studying sparse polynomials is to concentrate on the structure of a polynomial's monomials instead of its degree, as one can see, for instance, by comparing Bernstein–Kushnirenko theorem with Bézout's theorem. Research on sparse polynomials has also included work on algorithms whose running time grows as a function of the number of terms rather than on the degree, for problems including polynomial multiplication, division, root-finding algorithms, and polynomial greatest common divisors. Sparse polynomials have also been used in pure mathematics, especially in the study of Galois groups, because it has been easier to determine the Galois groups of certain families of sparse polynomials than it is for other polynomials. The algebraic varieties determined by sparse polynomials have a simple structure, which is also reflected in the structure of the solutions of certain related differential equations. Additionally, a sparse positivstellensatz exists for univariate sparse polynomials. It states that the non-negativity of a polynomial can be certified by SOS polynomials whose degree only depends on the number of monomials of the polynomial. Sparse polynomials often come up in sum or difference of powers equations. The sum of two cubes states that ( x + y ) ( x 2 − x y + y 2 ) = x 3 + y 3 {\displaystyle (x+y)(x^{2}-xy+y^{2})=x^{3}+y^{3}} . Here x 3 + y 3 {\displaystyle x^{3}+y^{3}} is a sparse polynomial, since out of the 16 {\displaystyle 16} possible terms, only 2 {\displaystyle 2} appear. Other examples include the identities ( x − y ) ∑ k = 0 N − 1 x k y N − 1 − k = x N − y N {\displaystyle (x-y)\sum _{k=0}^{N-1}x^{k}y^{N-1-k}=x^{N}-y^{N}} and also ( x + y ) ∑ k = 0 2 n ( − 1 ) k x k y 2 n − k = x 2 n + 1 + y 2 n + 1 , {\displaystyle (x+y)\sum _{k=0}^{2n}(-1)^{k}x^{k}y^{2n-k}=x^{2n+1}+y^{2n+1},} where the product of two polynomials give a sparse polynomial. The Bring–Jerrard normal form of a quintic, x 5 + p x + q , {\displaystyle x^{5}+px+q,} is also a sparse polynomial.

See also Askold Khovanskii, one of the main contributors to the theory of fewnomials.

References

Worked examples

Example 1 — a first encounter with Sparse polynomial

Start with the simplest possible case. Write down what Sparse polynomial claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Sparse polynomial 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 Sparse polynomial 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 Sparse polynomial

In research: Sparse polynomial appears in science 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 Sparse polynomial 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: Sparse polynomial is common in secondary-school and first-year university syllabi. It links to neighbouring topics Polynomial stubs, Polynomials, so understanding it makes those chapters shorter.
In everyday life: Look for Sparse polynomial 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 Sparse polynomial in 20 minutes

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

Frequently asked questions

What is Sparse polynomial in simple terms?

In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables would suggest. For example, x 10 + 3 x 3 + 1 {\displaystyle x^{10}+3x^{3}+1} is a sparse polynomial, as it is a trinomial with a degree of 10…

Why does Sparse polynomial matter?

Because it connects several science 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 Sparse polynomial?

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 Sparse polynomial.