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# Polynomial computations

## 8.1 Calculation of the value of a polynomial at the point

## 8.2 Factorization of polynomials. Bringing polynomials to the standard form.

## 8.3 Geometric progression. Summation of polynomial with respect to the variables

## 8.4 Groebner basis of polynomial ideal

## 8.5 Calculations in quotient ring of ideal

## 8.6 Solution of systems of nonlinear algebraic equations

## 8.7 Other polynomial functions

To calculate the value of a function at the point you must run value(f, [var1, var2, …, varn]), where $ f $ — is a polynomial in which the variables are replaced by the corresponding values of $ var1, var2, …, varn $.

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Polynomials are automatically brought to the standard form, which assumes, that the first written the leading monomials, and then younger. Recall that the order of variables in the ring determines seniority of variables.

To bring to the standard form of a polynomial can also be run expand(f), where $f$ — is a polynomial.

For factoring polynomials you must execute the command factor(f), where $f$ — it is a polynomial.

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For the summation of polynomial with respect to the variables we must run SumOfPol(f, [x, y], [x1, x2, y1, y2]), where $ f $ — a polynomial, $ x, y $ ~ — variables for summation, $ x1, x2 $ — range of summation over $ x $, $ y1, y2 $ — range of summation over $ y $.

If intervals of summation on all variables coincide then we can write SumOfPol(f, [x, y], [x1, x2]), where $ x1, x2 $ — summation interval for $ x $ and $ y $.

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To convert a polynomial using the formula of the sum of a geometric progression, you must run SearchOfProgression(f). This command searches for a geometric progression with the largest number of members in this polynomial. Then do it again for the remaining members, and so on. All the detected progression be written as $ S_n = b_1 (q ^ n-1) / (q-1) $, where $ S_n $ — sum of the first $ n $ members, $ b_1 $ — the first term of a geometric progression, $ q $ — the geometric ratio.

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The Groebner basis of polynomial ideal may be obtained due to the Bruno Buchberger algorithm groebnerB(). You can obtain the same basis using matrix algorithm, which similar to F4 algorithm groebnerB(). We use reverse lexicographical order. The order of the variables defined in the command SPACE,

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reduceByGB(f, [g_1, …, g_N]) function reduces polynomial $p$ with given set of polynomial $g_1, …, g_N$.

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In case when second argument isn't a reduced Groebner basis result depends on positions of polynomials in array: among all potential reductors the first one will be chosen.

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To obtain the solution of the polynomial system

$p_{1} = 0,$

$p_{2} = 0,$

...

$p_{N} = 0,$

use the command solveNAE(p_1, p_2, …, p_N).

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For polynomials in several variables (f, g), you can calculate GCD, LCM, a resultant (as the determinant of their Sylvester matrix), a discriminant:

** GCD** (f, g),

** LCM** (f, g),

** resultant** (f, g),

** discriminant** (f).

In this case, the main variable is the highest (last) variable, which is defined in the statement SPACE.

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