# Questions tagged [polynomials]

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1answer
178 views

### Efficient Shamir secret sharing reconstruction

Shamir's secret sharing scheme is a well known way to convert a secret into a polynomial and distribute points in this polynomial. Some of these points can then be regrouped to reconstruct the ...
1answer
167 views

### An upper bound over the number of bipolar orientations for a regular graph

Given a $k$-regular graph $G$, the number of acyclic orientations $Acy(G)$ is $\chi(-1)$ where $\chi$ is the chromatic polynomial of $G$. How many bipolar orientations does $G$ have? Is there an ...
2answers
171 views

### Root finding in [0,1]

I am interested in the problem of finding a real root of a polynomial equation $f(x)=0$ where $f(x)=\sum_{i=0}^n a_ix^i$. Is it possible to give a reduction, i.e, to compute a different polynomial $g$ ...
0answers
102 views

### The rank-polynomial of a graded poset

Let $P$ be a graded poset with rank function $r$. We may then define its rank-polynomial as: $R_P(q) = \sum_{x \in P} q^{r(x)}$. This definition can be applied to several interesting posets, for ...
1answer
143 views

### Where does randomness help when deciding algebraic geometry over $\mathbb{C}$?

If we have a single straight line program expressing a multivariate polynomial equation with integer coefficients, the Schwartz-Zippel lemma gives a simple randomized algorithm for deciding whether ...
3answers
1k views

1answer
352 views

1answer
228 views

### Bivariate low-degree polynomial testing of Polishchuk-Spielman

In the seminal paper of Polishchuk and Spielman where they give a construction of nearly linear sized $PCP$ for an $NP$ problem, one of the key ingredients is a low-degree test for bivariate ...
0answers
293 views

### Exponential-time factorization of polynomials

Let an explicit field be a field for which equality is decidable (in some standard model of computation). I am interested in the factorization of univariate polynomials over an explicit field. It is ...
0answers
75 views

### Verifying the minimum degree of a geometric predicate

Let $F = (f_1, \ldots, f_n)$ be a sequence of multivariate polynomials $f_i : \mathbb{R}^d \to \mathbb{R}$, and $g : \{0,1\}^n \to \{0,1\}$ a boolean function. Say the composition h(x)=g(f_1(x)>...
0answers
188 views

### Counting small terms in a determinant calculation over polynomials (counting spanning trees by weight)

I have a $n\times n$ matrix $A$. It's terms are $a_{ij}=-x^{w_{ij}}$ if $i\neq j$ and $a_{ii}=\sum_{j=0}^{n+1} x^{w_{ij}}$ on the diagonal. The matrix is symmetric as $w_{ij}=w_{ji}$. Numbers $w_{ij}$ ...
3answers
576 views

### Find the remainder of a large fixed polynomial when divided by a small unknown polynomial

Assume we operate in a finite field. We are given a large fixed polynomial p(x) (of, say, degree 1000) over this field. This polynomial is known beforehand and we are allowed to do computation using a ...
1answer
747 views

### Approximate degree of $\textrm{AC}^0$

EDIT (v2): Added a section at the end on what I know about the problem. EDIT (v3): Added discussion on threshold degree at the end. Question This question is mainly a reference request. I don't ...
1answer
264 views

### What are some results on algorithms that estimate polynomials over a given set of points?

There seem to be many randomized algorithms for polynomial identity testing, checking whether or not a given polynomial is zero. Are there any results of algorithms that do some sort of estimation of ...
2answers
366 views

### What's the bias of random polynomials with low degree over GF(2)?

I have a question concerning low-degree polynomials and probability: What is the (assyptotic behavior of the) probability that a random* polynomial, $p$, over GF(2), with degree $\le d$ and n ...
0answers
358 views

### Looking for an operator on polynomials

I have a small, self-contained, math question, whose motivation is from theoretical computer science (specifically, list decoding of algebraic codes, derivative/multiplicity codes, etc). I wonder ...
1answer
144 views

### Parallel algorithms to find the optimum of polynomials

Are there any non-trivial parallel algorithms to find the optimum (local or global) of polynomials? By trivial, I mean something which is an obvious application of a serial algorithm. For example, one ...
0answers
135 views

### Computing the Fourier-Walsh coefficients of an arithmetic circuit

Given an $f$-fan in arithmetic circuit of depth $d$ over $GF(q)$ (for some prime $q$) and the set of variables $(x_1,\ldots,x_n)$, what is the state of the art upper bound for calculating the Fourier-...
1answer
247 views

### Results in computer science via the Stepanov method

I recently attended a Workshop on Pseudorandomness in Chennai Mathematical Institute on Pseudorandomness. Venkat Guruswami made the following beautiful statement in passing during his talk (on coding ...
1answer
462 views

### Systematic studies of sum of quadratic polynomials squared

I'm wondering if there exists systematic studies of sums of quadratic forms squared, similar to the quadratic forms, which is practically reflected in eigenvalue decomposition (that has huge practical ...
12answers
2k views

### Gröbner bases in TCS?

Does anyone know of interesting applications of Gröbner bases to theoretical computer science? Gröbner bases are used to solve multi-variate polynomial equations, an NP-hard problem in general. I was ...
3answers
776 views

### Computing sum of sparse polynomials squared in O(n log n) time?

Suppose we have polynomials $p_1,...,p_m$ of degree at most $n$, $n>m$, such that the total number of nonzero coefficients is $n$ (i.e., the polynomials are sparse). I am interested in an efficient ...
0answers
203 views

### Is there an interpretation of the degree of the polynomial computed by the arithmetization of a boolean formula?

If I understand this correctly, arithmetizating a formula is a way of "interpolating" a polynomial in such a way that polynomial evaluated on 0's and 1's corresponds to "unsatisfiable" if it is a root,...
1answer
255 views

### Explicit polynomials in 1 variable with superlogarithmic circuit complexity lower bounds?

By counting arguments, one can show that there exist polynomials of degree n in 1 variable (i.e., something of the form $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0)$ which have circuit complexity n. ...
1answer
376 views

2answers
843 views

### Polynomial method for complexity results

Polynomial methods, say Combinatorial Nullstellensatz and Chevalley–Warning theorem are powerful tools in additive combinatorics. By representing a problem with proper polynomials, they can guarantee ...
1answer
598 views

### How to compute ROOK Polynomials for NxM Matrices [closed]

How to compute ROOK Polynomials for NxM Matrices for k objects ?
1answer
493 views

### What is the running time of taking a limit?

I'm interested in finding the running time(s) for determining mathematical limits. For instance, $\lim_{x \to 2} \frac{1}{x} = \frac{1}{2}$. I'd like to know more about algorithms for determining ...
6answers
2k views

### Alternative proofs of Schwartz–Zippel lemma

I'm only aware of two proofs of Schwartz–Zippel lemma. The first (more common) proof is described in the wikipedia entry. The second proof was discovered by Dana Moshkovitz. Are there any other ...
0answers
266 views

### Some problems involving polynomials of public and private variables over GF(2).

Suppose there are a set of low degree (less than some degree $z$) polynomials $P_0, P_1, ..., P_k$ each of which is defined over two types of variables, red variables ${v_r}_0, {v_r}_1, ..., {v_r}_n$ ...
1answer
2k views

### Multiplying n polynomials of degree 1

The problem is to compute the polynomial $(a_1 x + b_1) \times \cdots \times (a_n x + b_n)$. Assume that all coefficients fit in a machine word, i.e. can be manipulated in unit time. You can do \$O(n \...