Talk:Resultant

Multiplicity of roots in main definition

I think we have a problem with the definition. It says:

In mathematics, the resultant of two monic polynomials $P$ and $Q$ over a field $k$ is defined as the product

$\mathrm {res} (P,Q)=\prod _{(x,y):\,P(x)=0,\,Q(y)=0}(x-y),\,$

of the differences of their roots, where $x$ and $y$ take on values in an algebraic closure of $k$.

In other places, the resultant is defined using each root repeateadly according to its multiplicity. Here, each root is clearly used only once, disregarding multiplicity. For example, if $P$ and $Q$ have degrees $m$, and $n$ respectively, and their roots are $n+m$ different complex numbers, we have, according to the article, $Res(P^{2},Q)=Res(P,Q)$ (a product of $nm$ factors), which is different from the limit as $e->0$ of $Res(P(x-e)P(x),Q)$ (having $2nm$ factors), so this formula is not continuous and cannot depend polynomially on the coefficients of $P$ and $Q$. I'll change the definition to take into account the multiplicities, even though I'm not an expert in this field. [[[User:Marcosaedro|Marcosaedro]] (talk) 20:32, 2 December 2011 (UTC)]

Done. I also showed that a polynomial formula in terms of the coefficients is expected, from the fundamental theorem of symmetric polynomials. The comments on reduction of P mod Q may need revision, since I didn't understand them fully. [[[User:Marcosaedro|Marcosaedro]] (talk) 21:31, 2 December 2011 (UTC)]

Eliminant

Some clarification is needed about "eliminant". Although, I am quite an expert of the subject, I have never read "eliminant" with the meaning of "resultant". The term of "resultant" is standard since the work of McCaulay and the first editions of Van der Waerden, Moderne Algebra. As I have not checked if "eliminant" was used in the nineteenth century (Cayley, Bézout,...), I have not removed the mention of "eliminant", but I have added that it is no more in use.

Another thing has to be checked: It is possible that "eliminant" has been used with a meaning which is slightly different of that of "resultant": If p and q are 2 polynomials with coefficients in some commutative domain R. Then the resultant of p and q generates an ideal which has the same radical as $J=(p,q)\cap R$ but may be different. As J is usually called the elimination ideal, it is possible that some authors call "eliminant" a generator of J. This has to be checked. If true, this means that "eliminant" and "resultant" are not exactly the same thing.

D.Lazard (talk) 14:13, 7 May 2012 (UTC)

Style

I made some changes to remove most of the math tags and some of the bullet points. I found the article quite hard to read with all the huge Ps and Qs in it and believe that the bullet points (especially in the Applications section) did not help readability. At some places one should probably go back to the old math style though. Saraedum (talk) 14:23, 20 September 2012 (UTC)

It seems that I did some conflicting changes with User:D.Lazard that I was not warned about. Feel free to shift things around, I didn't mean to undo your changes. Saraedum (talk) 14:26, 20 September 2012 (UTC)

Lead

Before the recent edits by user:Saraedum the lead was not very good. But with these edits, the lead is a stub that does not satisfies the recommendations of MOS:LEAD and MOS:MATH. Specifically:

• An informal definition is needed; the previous formula of the old lead is better than nothing
• "Resultant" referring frequently to "multivariate" resultant, multivatiate resultants have to appear in the lead, at least for disambiguation purpose.
• The context provides by the old lead was poor, but was better than the empty present context.

I'll try to write an acceptable lead, please do not destroy it. D.Lazard (talk) 14:33, 20 September 2012 (UTC)

Sure, go ahead. I simply wanted to make it clear that most of the things that are written in the article are not about the multivariate resultant. Saraedum (talk) 14:41, 20 September 2012 (UTC)

I agree. My opinion is that Multivariate resultant is an important notion that needs a separate article. As it has never been written, we have, first, to mention it in the lead for people looking for it, then to expand the section devoted to it, and, when it becomes too long to split it in a separate article. Until that, I think that the present sentence in the lead id a good compromise. Remark that five months ago "multivariate resultant" did not appear anywhere in Wikipedia. D.Lazard (talk) 15:56, 20 September 2012 (UTC)

Error in the properties section

The text in the "characterizing properties" section is in error. Since my correction has been reverted twice, I'm bringing it to the talk page as I undo the most recent reversion.

The initial list of properties was (equivalent to):

• The resultant is multiplicative in each argument
• The resultant has a graded-symmetry
• The values the resultant has when the arguments are both constant or linear

This only characterizes the resultant in the special case that every polynomial splits into linears. If either argument has an irreducible nonlinear factor, these properties do not determine its value.

The most recent list of properties additionally lists the property that

• $\operatorname {res} (A,B)$ is a polynomial function (with integer coefficients) of the coefficients of A and B

This property is not actually satisfied by the resultant! For example, assuming a and c are nonzero, this implies

• $\operatorname {res} (ax+b,cx+d)=ad-bc$

However, substituting $a=0$ we arrive at a contradiction:

• $\operatorname {res} (b,cx+d)=b\neq -bc$

I understand several senses by which this statement is morally true, it is not actually true as stated.

While this statement can be fixed by introducing additional caveats and technical details, I strongly question the value of doing so at this specific place in the article. It would be better to make the initial, simple statement for algebraically closed fields, and state the more general characterization afterwards. It might even be ideal to delay that statement until the section on "Invariance by ring homomorphisms".

However, now that I read through it, the "Invariance by ring homomorphisms" section has similar errors. For example, let $R=\mathbb {Z} [t]$, $S=\mathbb {Z}$, and $\varphi (f)=f(0)$.

The article claims

• $\operatorname {res} (\varphi (tx^{2}+x+1),\varphi (tx^{2}+x+2))=0$

because the homomorphism kills both leading coefficients. However, we actually have

• $\operatorname {res} (\varphi (tx^{2}+x+1),\varphi (tx^{2}+x+2))=\operatorname {res} (x+1,x+2)=1\neq 0$

I'm not sure what precisely the correct statement is here; in any case I imagine it's complicated.

Hurkyl (talk) 18:04, 23 November 2017 (UTC)

Thank you for having carefully read the article. However, if you believe to find an error in a article for which you are not an expert, please tag the sentence with {{clarify|reason= your point}} or {{disputed inline|reason= your point}}, and discuss in the talk page. In fact it is possible that it is not an error but a badly written sentence that induces a confusion. Here, you have found two errors (a property ommitted in the characterizing properties, and another error that I have fixed). Also the writing may be confusing as the notation $\operatorname {res} _{d,e}(A,B)$ may be insufficiently used. But, in a any case, you must not restrict to algebraically closed fields properties that are widely used in the general case of integral domains. Here are some details on the points that you mention.

• "The resultant is a polynomial function (with integer coefficients) of the coefficients of A and B" In your examples, $ad-bc$ and b are both polynomials in a, b, c, d. Thus the assertion is true. The fact that you substitute values that change the degree, is explained in the section about ring homomorphisms.
• "$\operatorname {res} (b,cx+d)=b\neq -bc$": Here you make a substitution that changes the degrees. Also, using the notation defined in section notation:

$\operatorname {res} _{0,1}(b,cx+d)=b\neq -bc=\operatorname {res} _{1,1}(b,cx+d).$"

Maybe we should change "polynomial function of the coefficients" into "polynomial in the coefficients".

• "The article claims ...": It was an error, now fixed. D.Lazard (talk) 19:37, 23 November 2017 (UTC)

Come on

Is it really a good idea to denote the coefficient of $x^d$ $a_0$, against all mathematical conventions? — Preceding unsigned comment added by 89.138.175.87 (talk) 11:03, 12 July 2018 (UTC)

It is not a general convention, but it is common in elementary texts, to denote by $a_{i}$ the coefficient of $x^{i}.$ In theory of equations and number theory, the usual convention is to denote $a_{0}$ the leading coefficient. The reasons for that include the following: (a) It is not needed to know the degree for recognizing the leading coefficient, and this makes many formulas easier to understand (b) The index is the degree of the coefficient, when viewed as an elementary symmetric function of the roots. (c) When dealing with a monic polynomial of degree d, the coefficients are numbered from 1 to d, which is more comfortable than a numbering from 0 to d – 1. (d) For Euclidean division of polynomials, and many other algorithms one proceeds from the coefficient of the highest degree to the coefficients of lower degree. D.Lazard (talk) 14:26, 12 July 2018 (UTC)

On some properties listed in the article

Section moved from User talk:D.Lazard. Please do not modify it; for continuing the discussion, do it in next section. D.Lazard (talk) 08:38, 26 August 2020 (UTC)

Hello, I noticed that you were the original author of the subsection "Elimination Properties" in the Resultant article (this section appears for the first time in this revision made by you). Some of these statements seem unclear, incomplete, or perhaps even wrong to me. Regarding the first statement, that $I\cap R$ is principal, unless "a polynomial ring $R[x],$ where R is itself a polynomial ring over a field" means that $R[x]\cong k[x,y]$ for some field $k$, it seems like the ideal $I\cap R$ has no reason to be principal. For instance, if $R=k[y,z]$ and $A=z^{2}+xyz,B=yz$, then $I\cap R=(z^{2},yz)$ which isn't principal. Regarding the third bullet point of that subsection relating to some power of $r$ lying in the ideal generated by the resultant, if $R=k[y],A=xy,B=x^{2}y+y-1$, then $(A,B)\cap R=(y-1)$, while the resultant of these polynomials with respect to $x$ is $y^{2}(y-1)$, and no power of $y-1$ lies in the ideal generated by the resultant. I'd like to work towards a clearer and more correct version of the page - please let me know what you think. Kreiser math (talk) 07:54, 24 August 2020 (UTC)

Good catch. It seems that everything becomes correct if at least one polynomial is supposed to be monic in . But some more verifications are needed to be sure that with this restriction, the results are correct. Just now, I have not the time for that. D.Lazard (talk) 13:14, 24 August 2020 (UTC)

I agree that the third statement becomes true if we assume that one of the polynomials is monic and $R\cap I$ is principal (one can show that $V(res_{x}(A,B))=V(f,g)\cup V(R\cap I)$ where $f,g$ are the leading coefficients of $A,B$ respectively when considered as polynomials in $x$ - this follows from some of the theorems in chapter 3 of Cox, Little, and O'Shea), but I think the question of when $R\cap I$ is principal is somewhat complicated when $R$ is a polynomial ring in more than one variable. Did you intend for $R$ to be a polynomial ring in one variable, or any number of variables? Honestly, whether or not $R\cap I$ is principal and why is my biggest confusion here. Kreiser math (talk) 19:32, 24 August 2020 (UTC)

Let R be a polynomial ring in n variables, and $A,B\in R[x],$ such that at least one is monic. Their gcd is a monic polynomial. Therefore, if it is not 1, it is a polynomial of positive degree in x. This implies that the resultant and $(A,B)\cap R$ are both zero. So, we can suppose that the gcd is one.

A and B define hypersurfaces of dimension n. By unmixedness theorem, all primary components of the ideal $(A,B)$ have dimension n – 1. If $(A,B)\cap R$ is not principal, its dimension is at most n – 2. Thus the projection eliminating x reduces the dimension, which means that, for generic values of the variables in R, the fiber of the projection has a positive dimension, that is, A and B have an infinity of common zeros for the same values of the variables in R. This is impossible if one of the polynomials is monic.

I hope that this will convince you. However, I have a problem for the Wikipedia article: I do not know any reference for this result, and the proof seems too technical for the article. D.Lazard (talk) 08:20, 25 August 2020 (UTC)

Section "Elimination Properties"

I have fixed the section "Elimination Properties", by adding the condition that at least one of the polynomials must be monic, and adding Kreiser math's counterexamples (one has been generalized and simplified).

There is still to produce a source, or, if none is found, to add a proof. The above proof seems too technical, and I'll think for a simpler one that involves less algebraic geometry (because of the form of the third property, this should results more less directly from Hilbert's Nullstellensatz. D.Lazard (talk) 08:38, 26 August 2020 (UTC)

Finally, it seems that it is wrong that the elimination ideal is always principal, although I have no counterexample. So, I have rewritten the section for reducing it to the proved properties. Pinging to Kreiser math. D.Lazard (talk) 08:34, 3 September 2020 (UTC)