polynomials
Antihash test.  Codeforces
22 days ago by pent
 ThueMorse sequence
 nice paper: http://www.mii.lt/olympiads_in_informatics/pdf/INFOL119.pdf
In general, polynomial string hashing is a useful technique in construction of efficient string algorithms. One simply needs to remember to carefully select the modulus M and the variable of the polynomial p depending on the application. A good rule of thumb is to pick both values as prime numbers with M as large as possible so that no integer overflow occurs and p being at least the size of the alphabet.
2.2. Upper Bound on M
[stuff about 32 and 64bit integers]
2.3. Lower Bound on M
On the other side Mis bounded due to the wellknown birthday paradox: if we consider a collection of m keys with m ≥ 1.2√M then the chance of a collision to occur within this collection is at least 50% (assuming that the distribution of fingerprints is close to uniform on the set of all strings). Thus if the birthday paradox applies then one needs to choose M=ω(m^2)to have a fair chance to avoid a collision. However, one should note that not always the birthday paradox applies. As a benchmark consider the following two problems.
I generally prefer to use SchwartzZippel to reason about collision probabilities w/ this kind of thing, eg, https://people.eecs.berkeley.edu/~sinclair/cs271/n3.pdf.
A good way to get more accurate results: just use multiple primes and the Chinese remainder theorem to get as large an M as you need w/o going beyond 64bit arithmetic.
more on this: https://codeforces.com/blog/entry/60442
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via:nhaliday
 nice paper: http://www.mii.lt/olympiads_in_informatics/pdf/INFOL119.pdf
In general, polynomial string hashing is a useful technique in construction of efficient string algorithms. One simply needs to remember to carefully select the modulus M and the variable of the polynomial p depending on the application. A good rule of thumb is to pick both values as prime numbers with M as large as possible so that no integer overflow occurs and p being at least the size of the alphabet.
2.2. Upper Bound on M
[stuff about 32 and 64bit integers]
2.3. Lower Bound on M
On the other side Mis bounded due to the wellknown birthday paradox: if we consider a collection of m keys with m ≥ 1.2√M then the chance of a collision to occur within this collection is at least 50% (assuming that the distribution of fingerprints is close to uniform on the set of all strings). Thus if the birthday paradox applies then one needs to choose M=ω(m^2)to have a fair chance to avoid a collision. However, one should note that not always the birthday paradox applies. As a benchmark consider the following two problems.
I generally prefer to use SchwartzZippel to reason about collision probabilities w/ this kind of thing, eg, https://people.eecs.berkeley.edu/~sinclair/cs271/n3.pdf.
A good way to get more accurate results: just use multiple primes and the Chinese remainder theorem to get as large an M as you need w/o going beyond 64bit arithmetic.
more on this: https://codeforces.com/blog/entry/60442
22 days ago by pent
Operations on polynomials (on cpalgorithms)  Codeforces
8 weeks ago by nhaliday
https://stackoverflow.com/questions/44770632/fftdivisionforfastpolynomialdivision
links to good lecture notes: http://web.cs.iastate.edu/~cs577/handouts/polydivide.pdf
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links to good lecture notes: http://web.cs.iastate.edu/~cs577/handouts/polydivide.pdf
8 weeks ago by nhaliday
Color Me Polynomial  Quanta Magazine
combinatorics
math
polynomials
quantized
9 weeks ago by geetarista
Polynomials aren’t just exercises in abstraction. They’re good at illuminating structure in surprising places.
9 weeks ago by geetarista
Antihash test.  Codeforces
9 weeks ago by nhaliday
 ThueMorse sequence
 nice paper: http://www.mii.lt/olympiads_in_informatics/pdf/INFOL119.pdf
In general, polynomial string hashing is a useful technique in construction of efficient string algorithms. One simply needs to remember to carefully select the modulus M and the variable of the polynomial p depending on the application. A good rule of thumb is to pick both values as prime numbers with M as large as possible so that no integer overflow occurs and p being at least the size of the alphabet.
2.2. Upper Bound on M
[stuff about 32 and 64bit integers]
2.3. Lower Bound on M
On the other side Mis bounded due to the wellknown birthday paradox: if we consider a collection of m keys with m ≥ 1.2√M then the chance of a collision to occur within this collection is at least 50% (assuming that the distribution of fingerprints is close to uniform on the set of all strings). Thus if the birthday paradox applies then one needs to choose M=ω(m^2)to have a fair chance to avoid a collision. However, one should note that not always the birthday paradox applies. As a benchmark consider the following two problems.
I generally prefer to use SchwartzZippel to reason about collision probabilities w/ this kind of thing, eg, https://people.eecs.berkeley.edu/~sinclair/cs271/n3.pdf.
A good way to get more accurate results: just use multiple primes and the Chinese remainder theorem to get as large an M as you need w/o going beyond 64bit arithmetic.
more on this: https://codeforces.com/blog/entry/60442
oly
olyprogramming
gotchas
howto
hashing
algorithms
strings
random
bestpractices
counterexample
multi
pdf
papers
nibble
examples
fields
polynomials
lecturenotes
yoga
probability
estimate
magnitude
hacker
adversarial
CAS
lattice
discrete
 nice paper: http://www.mii.lt/olympiads_in_informatics/pdf/INFOL119.pdf
In general, polynomial string hashing is a useful technique in construction of efficient string algorithms. One simply needs to remember to carefully select the modulus M and the variable of the polynomial p depending on the application. A good rule of thumb is to pick both values as prime numbers with M as large as possible so that no integer overflow occurs and p being at least the size of the alphabet.
2.2. Upper Bound on M
[stuff about 32 and 64bit integers]
2.3. Lower Bound on M
On the other side Mis bounded due to the wellknown birthday paradox: if we consider a collection of m keys with m ≥ 1.2√M then the chance of a collision to occur within this collection is at least 50% (assuming that the distribution of fingerprints is close to uniform on the set of all strings). Thus if the birthday paradox applies then one needs to choose M=ω(m^2)to have a fair chance to avoid a collision. However, one should note that not always the birthday paradox applies. As a benchmark consider the following two problems.
I generally prefer to use SchwartzZippel to reason about collision probabilities w/ this kind of thing, eg, https://people.eecs.berkeley.edu/~sinclair/cs271/n3.pdf.
A good way to get more accurate results: just use multiple primes and the Chinese remainder theorem to get as large an M as you need w/o going beyond 64bit arithmetic.
more on this: https://codeforces.com/blog/entry/60442
9 weeks ago by nhaliday
Rational Sines of Rational Multiples of p
july 2019 by nhaliday
For which rational multiples of p is the sine rational? We have the three trivial cases
[0, pi/2, pi/6]
and we wish to show that these are essentially the only distinct rational sines of rational multiples of p.
The assertion about rational sines of rational multiples of p follows from two fundamental lemmas. The first is
Lemma 1: For any rational number q the value of sin(qp) is a root of a monic polynomial with integer coefficients.
[Pf uses some ideas unfamiliar to me: similarity parameter of Moebius (linear fraction) transformations, and finding a polynomial for a desired root by constructing a Moebius transformation with a finite period.]
...
Lemma 2: Any root of a monic polynomial f(x) with integer coefficients must either be an integer or irrational.
[Gauss's Lemma, cf DummitFoote.]
...
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[0, pi/2, pi/6]
and we wish to show that these are essentially the only distinct rational sines of rational multiples of p.
The assertion about rational sines of rational multiples of p follows from two fundamental lemmas. The first is
Lemma 1: For any rational number q the value of sin(qp) is a root of a monic polynomial with integer coefficients.
[Pf uses some ideas unfamiliar to me: similarity parameter of Moebius (linear fraction) transformations, and finding a polynomial for a desired root by constructing a Moebius transformation with a finite period.]
...
Lemma 2: Any root of a monic polynomial f(x) with integer coefficients must either be an integer or irrational.
[Gauss's Lemma, cf DummitFoote.]
...
july 2019 by nhaliday
Factorization of polynomials over finite fields  Wikipedia
july 2019 by nhaliday
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm. In practice, algorithms have been designed only for polynomials with coefficients in a finite field, in the field of rationals or in a finitely generated field extension of one of them.
All factorization algorithms, including the case of multivariate polynomials over the rational numbers, reduce the problem to this case; see polynomial factorization. It is also used for various applications of finite fields, such as coding theory (cyclic redundancy codes and BCH codes), cryptography (public key cryptography by the means of elliptic curves), and computational number theory.
As the reduction of the factorization of multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with one variable are considered in this article.
...
In the algorithms that follow, the complexities are expressed in terms of number of arithmetic operations in Fq, using classical algorithms for the arithmetic of polynomials.
[ed.: Interesting choice...]
...
Factoring algorithms
Many algorithms for factoring polynomials over finite fields include the following three stages:
Squarefree factorization
Distinctdegree factorization
Equaldegree factorization
An important exception is Berlekamp's algorithm, which combines stages 2 and 3.
Berlekamp's algorithm
Main article: Berlekamp's algorithm
The Berlekamp's algorithm is historically important as being the first factorization algorithm, which works well in practice. However, it contains a loop on the elements of the ground field, which implies that it is practicable only over small finite fields. For a fixed ground field, its time complexity is polynomial, but, for general ground fields, the complexity is exponential in the size of the ground field.
[ed.: This actually looks fairly implementable.]
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All factorization algorithms, including the case of multivariate polynomials over the rational numbers, reduce the problem to this case; see polynomial factorization. It is also used for various applications of finite fields, such as coding theory (cyclic redundancy codes and BCH codes), cryptography (public key cryptography by the means of elliptic curves), and computational number theory.
As the reduction of the factorization of multivariate polynomials to that of univariate polynomials does not have any specificity in the case of coefficients in a finite field, only polynomials with one variable are considered in this article.
...
In the algorithms that follow, the complexities are expressed in terms of number of arithmetic operations in Fq, using classical algorithms for the arithmetic of polynomials.
[ed.: Interesting choice...]
...
Factoring algorithms
Many algorithms for factoring polynomials over finite fields include the following three stages:
Squarefree factorization
Distinctdegree factorization
Equaldegree factorization
An important exception is Berlekamp's algorithm, which combines stages 2 and 3.
Berlekamp's algorithm
Main article: Berlekamp's algorithm
The Berlekamp's algorithm is historically important as being the first factorization algorithm, which works well in practice. However, it contains a loop on the elements of the ground field, which implies that it is practicable only over small finite fields. For a fixed ground field, its time complexity is polynomial, but, for general ground fields, the complexity is exponential in the size of the ground field.
[ed.: This actually looks fairly implementable.]
july 2019 by nhaliday
parts of an equation with polynomials  Google Search
may 2019 by neerajsinghvns
https://www.google.com/search?q=parts+of+an+equation+with+polynomials&source=lnms&tbm=isch&sa=X&ved=0ahUKEwinwqS634fiAhVGZN8KHRAaAWAQ_AUIDigB&biw=1920&bih=977 ;;;

https://www.mathsisfun.com/algebra/polynomials.html ;;;
Polynomial or NOT.
A polynomial.
NOT a polynomial.

tags: parts of an equation with polynomials  Google Search  polynomial needsEditing ;;;
parts
of
an
equation
with
polynomials

Google
Search

polynomial
needsEditing
math
algebra

https://www.mathsisfun.com/algebra/polynomials.html ;;;
Polynomial or NOT.
A polynomial.
NOT a polynomial.

tags: parts of an equation with polynomials  Google Search  polynomial needsEditing ;;;
may 2019 by neerajsinghvns
Addition and Subtraction of Polynomials ( Read )  Algebra  CK12 Foundation
may 2019 by neerajsinghvns
https://www.ck12.org/algebra/additionandsubtractionofpolynomials/lesson/AddingandSubtractingPolynomialsALGII/ ;;;

tags: Addition and Subtraction of Polynomials ( Read )  Algebra  CK12 Foundation  polynomial needsEditing ;;;
Addition
and
Subtraction
of
Polynomials
(
Read
)

Algebra
CK12
Foundation

polynomial
needsEditing
math

tags: Addition and Subtraction of Polynomials ( Read )  Algebra  CK12 Foundation  polynomial needsEditing ;;;
may 2019 by neerajsinghvns
[1812.02907] Caustics of Poncelet polygons and classical extremal polynomials
december 2018 by Vaguery
A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of d intervals on the real line and ellipsoidal billiards in ddimensional space. Even in the planar case, systematically studied in the present paper it leads to new results in characterizing n periodic trajectories vs. socalled n elliptic periodic trajectories, which are nperiodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell's equation. This new approach also sheds light on some classical results. In particular we connect search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for n=3,4,5,6 along with an effective search for caustics. As a byproduct, an intriguing connection between Cayle type conditions and discriminantly separable polynomials has been observed for all those small periods.
dynamicalsystems
billiards
polynomials
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ratherinteresting
december 2018 by Vaguery
Twitter
november 2018 by nigeljames
Life of travelling consultant means I get to do remote maths homework help... #pythagoras #foil #maths #polynomials
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from twitter
november 2018 by nigeljames
Programmer's guide to polynomials and splines
february 2018 by Tafkas
So you are a programmer. Why would you want to know about polynomials? One reason would be that this is the geometrical clay we can easily make different things of.
tutorial
polynomials
february 2018 by Tafkas