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Questions tagged [polynomials]

133 questions
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Find a polynomial in two or three queries

Black box of $f(x)$ means I can evaluate the polynomial $f(x)$ at any point. Input: A black box of monic polynomial $f(x) \in\mathbb{Z}^+[x]$ of degree $d$. Output: The $d$ coefficients of polynomial $f(x)$. My algorithm: let $$f(x) = x^{d} +…
Complexity
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Is there a complexity viewpoint of Galois' theorem?

Galois's theorem effectively says that one cannot express the roots of a polynomial of degree >= 5 using rational functions of coefficients and radicals - can't this be read to be saying that given a polynomial there is no deterministic algorithm…
user6818
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2 answers

What is the most efficient algorithm to compute polynomial coefficients from its roots?

Given $n$ roots, $x_1, x_2, \dotsc, x_n$, the corresponding monic polynomial is $$y = (x-x_1)(x-x_2)\dotsm(x-x_n) = \prod_{i}^n (x - x_i)$$ To get the coefficients, i.e., $y = \sum_{i}^n a_i x^i$, a straightforward expansion requires $O…
xucheng
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11
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1 answer

Multi-point evaluations of a polynomial mod p

Given a polynomial of degree $n$ modulo a prime number $p$, I want to evaluate that polynomial at multiple values of the variable $x$, what is the best way to do this? I tried using Berlekamp's algorithm for factorization but it takes $O(n^3)$ just…
imanimefn
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9
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1 answer

Algorithm for multiplying multivariate polynomials

Let $R$ be a commutative ring. Let $f(x_1, \dots, x_n), g(x_1, \dots, x_n)$ be two multidimensional polynomials in $R$ with maximal total degree $\delta$. How fast can we compute the product of $f$ and $g$, i.e. the resulting coefficients of each…
chtenb
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8
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Efficient algorithm to translate/shift polynomials

I have a polynomial $P(x)$, and given some constant $d$, I need to find the polynomial $P(x+d)$. For example, if $P(x)=x^2$ and $d=1$, then the result would be $P(x+1)=(x+1)^2=x^2+2x+1$ (with the coefficients stored in an array/vector). I know the…
tmwilliamlin168
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8
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Understanding Intel's algorithm for reducing a polynomial modulo an irreducible polynomial

I'm reading this Intel white paper on carry-less multiplication. It describes multiplication of polynomials in $\text{GF}(2^n)$. On a high level, this is performed in two steps: (1) multiplication of polynomials over $\text{GF}(2)$, and (2) reducing…
Gideon
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7
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Finding constants C and k for big-O of fraction of polynomials

I am a teaching assistant on a course for computer science students where we recently talked about big-O notation. For this course I would like to teach the students a general method for finding the constants $C$ and $k$ in the definition of…
mrp
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6
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1 answer

Complexity of polynomial interpolation

Polynomial Interpolation in the general case is $O(n^2)$ time complexity, but it can be done better in particular situations. For instance, when the polynomial can be evaluated at the complex roots of unity, it can be interpolated in $O(n\log n)$…
Vadim Pelyushenko
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6
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4 answers

Converting Polynomials into Binary form

How can a polynomial such as $x^3 + 1$ be converted to its binary form of $1001$. Likewise, $10100001 = x^7 + x^5 + 1.$
M-R
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6
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Check if a given polynomial is primitive

I try to estimate error detection capabilities of arbitrary CRC polynomials. One important criteria is if a given polynomial is primitive. So I need an algorithm to check that. My goal is to write a C or C++ routine. Unfortunately I only found…
Silicomancer
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6
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4 answers

Calculating the number of multiplications necessary to evaluate a polynomial

I was watching a lecture and got confused over a slide. This is what it says: Consider a polynomial - first representation $$P = 2 + 4x^{3} + 8x^{6} + 7x^{25} + 6x^{99}$$ The space complexity is 100 memory locations. The time complexity…
Shashi
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6
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Fast polynomial calculation over $\mathbb{Z}_{487}$

Given a polynomial $a(x)$ of degree at most $242$ over $\mathbb{Z}_{487}$, I'd like to choose distinct values $x_0, x_1, . . . , x_{242} ∈ \mathbb{Z}_{487}$, such that I'll be able to calculate $a(x_j )$ for all $j = 0, . . . , 242$ by calculating…
Eric_
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Testing whether a determinant polynomial is identically zero

Suppose we are given matrices $A_1, \ldots, A_k$ which are $n \times n$ matrices with rational entries and are asked to determine whether the polynomial ${\rm det}(\alpha_1 A_1 + \alpha_2 A_2 + \cdots + \alpha_k A_k)$ is identically zero. How can we…
Jean Michelle
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Lower bound of degree of polynomial approximating parity

Let $\text{MOD}_2 : \{0,1\}^n \rightarrow \{0,1\}$ be a parity function where $$\text{MOD}_2(x_1,\dots,x_n) = \sum_i x_i \bmod 2$$ It is known [See e.g. Lemma 5 of this lecture note] that any polynomial $f(x_1,\dots,x_n) \in…
eig
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