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To understand the stuff of the present page, it would be desirable to remind some prerequisites in polynomial theory and determinant formalism.

Denote by any of the sets or .

For the polynomials and при consider the square matrix of the order

(the entries above and , and below and equal to zero). The expression

is called **the resultant**1) (**in the Sylvester form**) of the polynomials and .

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In the majority of textbooks, the definition of the resultant is presented in a slightly different manner: the rows of the coefficients of the polynomial are reordered in such a way that form the staircase ri sing up from the lower rigt corner. Thus, for instance,

Such a modification gives one an economy in the sign involved into the definition. Nevertheless, due to some reasons soon to be explained ☟ BELOW, it is more convenient for me to utilize the above presented form of the matrix .

Ex

**Example.**

Th

**Theorem.** *Denote the zeros of polynomial* *by* ,* and the zeros of polynomial * *by2) *. *One has*

=>

=>

if and only if the polynomials and possess a common zero.

Ex

**Example.** Find all the values for the parameter for which the polynomials
and possess a common zero.

**Solution.** Compute the determinant with the aid of elementary transformation of its rows

(the first row is subtracted from the last one, while the second row from the last-but-one), expand the resulting determinant by the last column:

(the first row multiplied by is subtracted from the last one), expand the determinant by the first column:

(the first row is added to the second one, the first row multiplied by is added to the third one), expand the determinant by the last column:

**Answer.** .

**Check.** .

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Find all the values for the parameter for which the polynomials

possess a multiple zero.

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For the particular case , the resultant is transformed into discriminant.

!

The main application of the resultant ☞ elimination of variables in a system of polynomial equations .

1. If and , then

2.

3.

The last equality is valid under an additional assumption that

4. If and , and then

By definition, resultant is a polynomial over with respect to the coefficients of polynomials and :

Its degree equals . Treated with respect to , the resultant is a homogeneous polynomial of the degree ; treated with respect to the resultant is a homogeneous polynomial of the degree .

Th

**Theorem.** *If polynomials* and *possess a unique common root* *then*

*for any* .

**Proof.** Consider the case . Differentiate the equality for the resultant

with respect to . One gets:

If polynomials and possess a unique common root then the last equality is transformed into

and the product does not vanish. Since the equality is valid for any specializations of , one has

and the other equality from the theorem is proved similarly. ♦

Consider the matrix from the definition of the resultant and delete its first and last columns as well as the first and the last rows. We get a square matrix of the order . Its determinant

is called the **first subresultant** of the resultant .

Th

**Theorem.** *For the polynomials* *and* *to possess a unique common zero it is necessary and sufficient that the following conditions be satisfied:*

=>

Under the conditions of the previous theorem, the single common zero of and can be expressed as a rational function of the coefficients of these polynomials:

Here the denotes the matrix obtained from by deleting its first and last rows as well as its first and its *last but one* column.

Ex

**Example.** Under the conditions of the theorem, the single common zero of the polynomials

and

() can be expressed by the formula:

♦

The determinant of the matrix obtained from the matrix by deleting its first columns and its last columns, its first
rows and its last rows, is called the **th subresultant** of the resultant
of the polynomials and ; we will denote it as . For simplicity, we term by the
**zero subresultant** the determinant of the matrix :

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**Example.** For the polynomials and one has:

♦

Th

**Theorem.** *Polynomials* *and* *possess the greatest common divisor of the degree* , *it is necessary and sufficient the fulfillment of the following conditions*

=>

Under the conditions of the previous theorem, the greatest common divisor for the polynomials and can be represented in the form

Here stands for the matrix obtained from the subresultant by the replacement of its last column with

(we set here for and for ).

Ex

**Example.** Find the greatest common divisor for the polynomials

**Solution.** Omitting the intermediate computations, we present the final result:

Therefore, is of the degree . For its computation, compose the determinant by replacing the last column of the subresultant :

**Answer.** .

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For the particular case , the subresultants transform into subdiscriminants.

Th

**Theorem.** *Let* * be an arbitrary polynomial. Compute the polynomial from the square matrix * :

*Then*

*where* *is the resultant of the polynomials* and .

**Problem.** For the polynomials from :

(); denote (a priori not known)
roots of . Construct the polynomial with the roots .
Process of computation of such a polynomial is known as the **Tschirnhaus transformation**
of the polynomial .

Th

**Theorem.** *There exists a unique normalized polynomial* *of the degree* *solving the stated problem, namely*

*here the resultant is computed for polynomials treated in the variable* . *Coefficients of* *depend rationally on those of
* and .

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**Example.** *Find the Tschirnhaus transformation*
*for polynomial* .

**Solution.**

♦

Generalizations of the problem are as follows

**Problem.** For the polynomial possessing the roots , construct the polynomial with the roots

**(a)** ;

**(b)** ;

**(c )** ;

here .

The polynomial with complex coefficients is called **stable**, if all its zeros satisfy the condition .

The notion of *stability* is a crucial one in the Optimal Control Theory.
The next result gives one of the most popular criterion for stability.

Th

**Theorem [ Liénard, Chipart]** [3]. *The polynomial
* *with real coefficients and* *is stable if and only if the following conditions are satisfied:*

**(a)** *all the coefficients* *are positive*;

**(b)** *all the subresultants of the resultant*

*are positive.*

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**Example.**
Condition **(b)** for :

and for :

and .
The last inequality follows from **(a)**.

**Problem.** Let and be polynomials in and with complex coefficients. Solve the system of equations

i.e. find all the pairs with , such that their substitution in every equation yield the true equalities.

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For the case of linear polynomials and the solution to the problem leads to the idea of Gaussian elimination.

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If the coefficients of the considered polynomials are real then the geometric interpretation of the stated problem is as follows. Every equation or defines in the -plane some algebraic curve. Then the problem of finding the *real* solutions for the system is that of detecting the intersection points for these curves. A particular case of the intersection might be tangency of the curves.

Expand the polynomials and in decreasing powers of the variables and represent the result as sums of homogeneous polynomials (forms):

Set the following assumption concerning the leading coefficients of the forms of the highest orders:

Assumption. Let

The pair with is a solution to the sustem if and only if the polynomials and possess a common zero , and, therefore, due to the main feature of the resultant:

Expand and in decreasing powers of

(here due to Assumpion ; and , ) and compute the Sylvester determinant:

The expression is a polynomial in and, by construction, it has the coefficients from the same set as those of and . For the pair to be a solution for the system, it is necessary that be the zero of .

Polynomial , i.e. the resultant of and treated as polynomials in

is known as the **eliminant**3) of the system of equations **with respect to the variable** . In a similar manner, the second eliminant for the system is defined:

With the aid of eliminant, one can reduce the solution of the system in two variables to the solution a univariate equation
: (or ). It is said that the other variable *is eliminated*. The corresponding branch of classical algebra is known as **Elimination Theory**.

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For simplicity, I did not care in the above example (and will not care in the foregoing ones) about the sign in the expressions of the both eliminants.

Ex

**Example.** Solve the system of equations

**Solution.** Expand polynomials of the system into powers of :

and compute the eliminant:

Find its zeros: .

Thus, the -components (coordinates) of solutions are found. How to find the -components? One might construct the other eliminant , find then its zeros, compose all the possible pairs of the zeros and , substitute them into and in order to establish whether we get zeros or not. Or, alternatively, substitute the found value into any of the initial equations: , solve the obtained with respect to , and substitute the resulting pairs into . At least one of them has to satisfy the relation . In the present example, any of the suggested approaches leads one to the correct

**Answer.** .

However, it is well known that generically zeros of a polynomial cannot be expressed — not even in the integers — but also in «good» functions of polynomial coefficients (say, for instance, in radicals). Therefore, one might expect that the zeros of the eliminant can be evaluated only approximately. Then the outlined in the previous example algorithm for selecting an appropriate pair for the variable values becomes inadequate: the eqaulity should be treated as an approximate one and the roundoff error might cause the wrong conclusion.

Ex

**Example.** Solve the system of equations

**Solution.** The eliminant

has zeros

some of which are close enough. Note that for any zero of the eliminant , the polynomials and treated as polynomials in must have a common zero. This common zero, in case of its uniqueness, can be expressed as a rational function in the coefficients of these polynomials with the aid of subresultants. Formula

for any zero of the eliminant yields the value for such that the obtained pair happens to be a solution to the given system of equations.

**Answer.**

**Resume.** Generically, the system of polynomial equations can be reduced to an equivalent one (i.e. possessing the same set of solutions) in the form:

Here are polynomials in .

How many solutions has the system of equations ? — It is evident that this number coincide with the degree of the eliminant:

Th

**Theorem [Bézout].** *Under the*
assumption
*from the previous section, one has, generically, *

**Proof** (taken from [5]) will be illuminated by the case and .

Here

for

The leading term of is formed from the leading terms of the determinant entries. Extract them:

Hence,

and we need to extract the power of from the first determinant. Let us carry out this with the aid of procedure which can be easily extended to the case of arbitray polynomials and . Multiply the second and the fourth rows by , while the third one by :

Extract from the second column the common divisor of its entries, from the third column — , from the fourth — , from the fifth — :

!

and pay attention that the obtained determinant has the form of the resultant in the Sylvester form for some univariate polynomials:

As for arbitrary and , one gets

and similarly

for

and

♦

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Prove that the leading coefficients of and coincide up to a sign.

**Hint.** V. the property
4
☞
HERE.

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Thus, we have just clarified the sence of the word «generically» from the theorem statement: it has the meaning that
the number differs from zero. it is necessary to emphasize that this number depends *only* on the coefficients of the leading forms in the expansions of and .

[1]. **Bôcher M.** *Introduction to Higher Algebra*. NY. Macmillan, 1907

[2]. **Jury E.I.** *Inners and Stability of Dynamic Systems.* J.Wiley & Sons, New York, NY, 1974.

[3]. **Kronecker L.** *Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen.* Werke. Bd. 2. 1897. 113-192, Teubner, Leipzig

[4]. **Gantmacher F.R.** *The Theory of Matrices.* Chelsea, New York, 1959

[5]. **Brill A.** *Vorlesungen über ebene algebraischen Kurven und algebraische
Funktionen.* Braunschweig. Vieweg. 1925

[6]. **Kalinina E.A., Uteshev A.Yu.** Elimination Theory (in Russian) SPb, Nii khimii, 2002

[7]. **Утешев А.Ю., Калинина Е.А.** *Лекции по высшей алгебре. Часть II.* Учеб. пособие. СПб. «СОЛО». 2007.

[8]. **Kalinina E.A., Uteshev A.Yu.** *Determination of the Number of Roots of a Polynominal Lying in a Given Algebraic Domain.* Linear Algebra Appl. 1993. V.185, P.61-81.