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Questions tagged [algebraic-curves]

An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

If $K$ is a field, then an algebraic curve is an equation $f(X,Y)=0$ where $f(X,Y)$ is a polynomial with coefficients in the field. In other words, it is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

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Problem Fulton Algebraic Curves

I have to do the problem 2.50, I assume the problem 2.30 done. So I did the following: b) I know that (t)=m that is equal to ${t,t^2,...}$ is the generator so I have that $R/m^n={u(1,t,t^2,...)/u(t^n,...)}$ where u is the unity, so $R/m^n$ has n…
Kc2
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Find X given Y in a cubic function.

Having asked this question on the math overflow boards one of the contributors suggested this may be a more appropriate forum. I have a cubic function in the form: $$y = ax^3 + bx^2 + cx + d$$ ...(where a, b, c and d are all known constants e.g…
John
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Does the singularity of a projective plane curve depend upon the system of local co-ordinates?

Let $F$ be a projective plane curve on $P_2(k)$ where $k$ is a field. Then, we introduce a system of local co-ordinates to convert this projective curve into an affine curve i.e. we take a point $(x_0,y_0,z_0)$ in $P_2(k)$ and take a projective…
DpS
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The relationship between ramification index and "degree of maps between algebraic curves"

This problem is Exercise II.2.3. in Silverman's 'Arithmetic of elliptic curves' . Consider in the field $\mathbb{C}$. We need to verify directly that \begin{equation} \sum_{P\in \phi^{-1}(Q)}e_{\phi}(P)=deg(\phi) \text{ for all }Q\in…
Edelweiss Ntu
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Solving a curve of fifths

I have five questions (A to E) used in a scorecard, all are currently ranked 0 or 1 meaning if all are answered 1, the total score possible is 5. I want the total of all to be 100 where the increments are mathematically stepped. The sum can be no…
Capitalmind
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5th order Polynomial not accurate enough?

I have a data plot XY that goes from (X 0-127, Y -70.0 - 6.0 db) Im using the 5th order polynomial function from plotting this data on this site [http://www.zizhujy.com/en-us/Plotter][1] However, its not accurate at all at the top end (comes out at…
Ke.
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Easy question about coordinate rings

Let $C=V(x^2+y^2-1)$ be an affine algebraic curve. In an online course the professor said $\varphi=\frac{x-1}{y}\in A(C)$, but he didn't explain why. I would like to know how he gets this function and why it's in the coordinate ring of the curve…
user42912
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How to derive a cubic equation $ax^3+bx^2+cx+d =y$ from $x$ and $y$.

Please let me show how to derive a cubic equation form $ax^3+bx^2+cx+d =y$ by using a set of $x$ and $y$ data. Simply the outline of the cubic equation derivation... Thank you
rrcmks
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Why/how does a type I (dividing) curve impart canonical orientations on its real ovals by virtue of its complex structure in $ℂ^2$

I've been unable to get over the hurdle of understanding this concept. However, I'm willing to bet that once I finally understand it will all seem obvious. I do understand that the real part of the curve divides the complexification of the curve…
Simon M
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Using the image of a line in a discrete valuation ring (DVR) as a uniformizing parameter (uniformizer)

Let $F$ be the zero locus of $x^2-y$ The origin is a simple point of $F$ Thus, the set of rational functions of $x$ and $y$ on $F$ defined at the origin is a discrete valuation ring $R$ Therefore, there exists an irreducible element (uniformizing…
Simon M
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Why isn't the zero locus of $y^3-x^2-y$ the union of the respective zero loci of 3 expressions of the form $y-f(x)$?

Let $C(F(x,y))$ denote the zero locus of $F(x,y)$ Let $A=${$(x,y) : y\ge\frac{2\sqrt{3}}{3}$} Let $B=${$(x,y) : 0\le y\le\frac{2\sqrt{3}}{3}$} Let $C=${$(x,y) : -\frac{\sqrt{3}}{3}\le y\le0$} Let $D=${$(x,y) : y\le-\frac{\sqrt{3}}{3}$} It can…
Simon M
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Real part of a rational curve

Let $C\subset\mathbb{P}^2$ be a rational curve of degree $d$. Let $k$ be the number of real places at infinity of $C$ Now, does that mean that there exists a rational parametrization of $C$ in which the denominator can be chosen to have at most $k$…
user382144
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Polynomial parametrization and degree nominator of parametrization of a rational curve

Let $\mathcal{C}\subset\mathbb{R}^2$ given by $f(x,y)=0$ be a rational curve. Then, there exists a rational parametrization $\phi(t)\,:\mathbb{R}\rightarrow \mathbb{R}²$. Under which circumstances is this parametrization polynomial? For example, if…
user382144
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How to compute the intersection number at singularities?

I'm studying algebraic curves and compact riemann surfaces. I read several related nice textbooks such as Kirwan, Griffiths and Miranda. I met such a concrete example in Griffiths' book: on page 206, test problem 5, I have to compute the…
youknowwho
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A way to study families of algebraic curves

I was looking for a way to study rational points on a family of curves instead of only one at a time, is there any?
Guilherme Gondin
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