View source for Resultant

''of two polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r0816601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r0816602.png" />''

The element of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r0816603.png" /> defined by the formula:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r0816604.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r0816605.png" /> is the splitting field of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r0816606.png" /> (cf. [[Splitting field of a polynomial|Splitting field of a polynomial]]), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r0816607.png" /> are the roots (cf. [[Root|Root]]) of the polynomials

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r0816608.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r0816609.png" /></td> </tr></table>

respectively. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166010.png" />, then the polynomials have a common root if and only if the resultant equals zero. The following equality holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166011.png" /></td> </tr></table>

The resultant can be written in either of the following ways:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

The expressions (1)–(3) are inconvenient for computing the resultant, since they contain the roots of the polynomials. Using the coefficients of the polynomials, the resultant can be expressed in the form of the following [[Determinant|determinant]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166014.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

This determinant contains in the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166016.png" /> rows the coefficients of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166017.png" />, and in the last <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166018.png" /> rows the coefficients of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166019.png" />, and in the free spaces there are zeros.

The resultant of two polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166021.png" /> with numerical coefficients can be represented in the form of a determinant of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166022.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166023.png" />). For this one has to find the remainders from the division of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166024.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166026.png" />. Let these be

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166027.png" /></td> </tr></table>

Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166028.png" /></td> </tr></table>

The [[Discriminant|discriminant]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166029.png" /> of the polynomial

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166030.png" /></td> </tr></table>

can be expressed by the resultant of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166031.png" /> and its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166032.png" /> in the following way:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166033.png" /></td> </tr></table>

==Application to solving a system of equations.==
Let there be given a system of two algebraic equations with coefficients from a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166034.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

The polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166037.png" /> are written as polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166038.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166039.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166040.png" /></td> </tr></table>

and according to formula (4) the resultant of these polynomials (as polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166041.png" />) is calculated. This yields a polynomial that depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166042.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166043.png" /></td> </tr></table>

One says that the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166044.png" /> is obtained by eliminating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166045.png" /> from the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166047.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166049.png" /> is a solution of the system (5), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166050.png" />, and, conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166051.png" />, then either the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166052.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166053.png" /> have a common root (which must be looked for among the roots of their greatest common divisor), or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166054.png" />. Solving system (5) is thereby reduced to the computation of the roots of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166055.png" /> and of the common roots of the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081660/r08166057.png" /> in one indeterminate.

By analogy, systems of equations with any number of unknowns can be solved; however, this problem leads to extremely cumbersome calculations (see also [[Elimination theory|Elimination theory]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) {{MR|0945393}} {{MR|0926059}} {{MR|0778202}} {{MR|0759341}} {{MR|0628003}} {{MR|0384363}} {{ZBL|0237.13001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.Ya. Okunev, "Higher algebra" , Moscow-Leningrad (1979) (In Russian) {{MR|}} {{ZBL|0154.26401}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German) {{MR|1541390}} {{ZBL|1032.00002}} {{ZBL|1032.00001}} {{ZBL|0903.01009}} {{ZBL|0781.12003}} {{ZBL|0781.12002}} {{ZBL|0724.12002}} {{ZBL|0724.12001}} {{ZBL|0569.01001}} {{ZBL|0534.01001}} {{ZBL|0997.00502}} {{ZBL|0997.00501}} {{ZBL|0316.22001}} {{ZBL|0297.01014}} {{ZBL|0221.12001}} {{ZBL|0192.33002}} {{ZBL|0137.25403}} {{ZBL|0136.24505}} {{ZBL|0087.25903}} {{ZBL|0192.33001}} {{ZBL|0067.00502}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''1–3''' , Cambridge Univ. Press (1947–1954) {{MR|1288307}} {{MR|1288306}} {{MR|1288305}} {{MR|0061846}} {{MR|0048065}} {{MR|0028055}} {{ZBL|0796.14002}} {{ZBL|0796.14003}} {{ZBL|0796.14001}} {{ZBL|0157.27502}} {{ZBL|0157.27501}} {{ZBL|0055.38705}} {{ZBL|0048.14502}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1984) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR></table>