Sparse polynomial
In mathematics, a sparse polynomial (also lacunary polynomial[1] or fewnomial)[2] is a polynomial that has far fewer terms than its degree and number of variables would suggest. For example, x10 + 3x3 − 1 is a sparse polynomial as it is a trinomial with a degree of 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
division,[6] root-finding algorithms,[7] and polynomial greatest common divisors.[8] 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.[9]
The
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.[10]
Sparse polynomials oftentimes come up in sum or difference of powers equations. The sum of two cubes states that (a + b)(a2 − ab + b2) = a3 + b3. a3 + b3, here, is a sparse polynomial.
References
See also
- Askold Khovanskii, one of the main contributors to the theory of fewnomials.
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