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Identity source: mathshistory.st-andrews.ac.uk/Biographies/Sylvester/

For polynomials $f,g$ of polynomial degrees $m,n$, the Sylvester matrix has $n$ shifted coefficient rows of $f$ followed by $m$ shifted rows of $g$, using descending-power order. It is a square matrix of size $m+n$. Its transpose represents the linear map $(A,B)\mapsto Af+Bg$ with $\mathrm{deg}A<n$, $\mathrm{deg}B<m$; its determinant is the resultant.

For an original parameter example, choose $f(x)=x^2-1$ and $g_t(x)=x-t$. ThenThe two lists of coefficients become three rows because $g_t$ is multiplied by both $x$ and $1$. Thus the common-root parameter values are $t=-1$ and $t=1$. At $t=1/2$, the resultant is $-3/4$, so the two polynomials have no common root.

One can verify the certificate without any determinant expansion:For $t\ne \pm 1$, division by $t^2-1$ gives a Bézout's identity for polynomials. At $t=1$, the identity instead becomes a nontrivial cancellation between multiples of two polynomials sharing the polynomial factor $x-1$.

The general proof needs only a short linear algebra outline. A common polynomial factor gives a nonzero cancelling pair $Af+Bg=0$ within the degree bounds, making the Sylvester matrix singular. Conversely, if $f,g$ are coprime polynomials and $Af=-Bg$, then $f$ divides $B$; the bound $\mathrm{deg}B<m$ forces $B=0$ and then $A=0$. Thus the linear map has zero kernel of a linear map exactly in the coprime case. Over an algebraic closure, a nonconstant common polynomial factor is equivalent to a common root.

In 1840, James Joseph Sylvester described forming the shifted coefficient rows and taking a signed sum of products selected by permutations. In modern notation, this is the Sylvester matrix and its determinant. The construction turns the nonlinear common-root of a polynomial question into linear algebra. His paper's word derivatives concerns elimination, rather than the formal derivative used below.^[ref]

Collected papers by James Joseph Sylvester, edited by Henry Frederick Baker, credited as H. F. Baker in the prefatory note dated April 1904. Cambridge University Press published Volume I in 1904. The collection date is distinct from the original dates of its individual papers. Internet Archive hosts the inspected digitized volume and OCR text.

Digitized volume: archive.org/details/collectedmathem01sylvrich

Inspected OCR: archive.org/stream/collectedmathem01sylvrich/collectedmathem01sylvrich_djvu.txt

Paper 9 by James Joseph Sylvester, reprinted on pp. 54–57 in The Collected Mathematical Papers of James Joseph Sylvester, Volume I. The reprint identifies the original publication as Philosophical Magazine XVI (1840), 132–135. Henry Frederick Baker edited the collection; he is not credited as author of this paper.

Inspected collection OCR: archive.org/stream/collectedmathem01sylvrich/collectedmathem01sylvrich_djvu.txt

Cited passages: paper 9, printed pp. 54–55, rules 1–4, shifted coefficient rows and the permutation sign sum. The historical word derivatives in the title is not used here to mean differentiation in calculus. The title page and prefatory note establish the separate 1904 collection date.