⎈ captn's log (PDF)

23.8.2021 : Upper bound for the intersection multiplicity

Looks like the bound $z_0 \leq \frac{mD}{t} + m +n$ for might possibly work? Hopefully? ($D=\max\{ \deg(F), \deg(G) \}$).

EDIT 26.8.: it didn't :(

13.8.2021 : Upper bound for the intersection multiplicity (PDF)

Since the pyramids were a dissapointment :(

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11.8.2021 : ssssssssssssssss

14.7.2021 : Continuation of the case of difference between and Blow-up (PDF)

This is basically a continuation of the post from 17.5.2021. The graphs of the branches $C_1$ and $C_2$ mentioned below can be seen here: [LINK TO DESMOS].

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8.7.2021: A little Corollary (PDF)

This basically says that if the intersection multiplicity is greater than $mn+t$, then it is increased at least by some number of branches. But I think this can be improved a little, it's more like a partial result.

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18.6.2021: Fixing the mistake from the Project of Disseration (PDF)

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2.6.2021: I don’t like Omega symbol anymore (i never liked it actually >:-( )

It’s too unspecific. I want to use some kind of spiral. This one will do for now: (ale chcelo by to nejaku usadlejsiu, tato ide moc do vysky). Or maybe something like this could do: (v pripade nudze mozno)? I don’t know.

17.05.2021 : Why is the Omega sequence different from the Blow-up (PDF)

Both blowup and our new decomposition (for pair of curves) provide us with a certain sequence of numbers. Sum of this sequence is equal to intersection multiplicity. These sequences are not equal in a sense that two cases having two identical blowup sequences does not imply having identical Omega sequences and vice versa.

Examples of this are below.

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Of course, there might be some correlation in certain types of intersections, it just looks like there is none in the general case.

PLOTS OF THE INTERSECTIONS IN THE EXAMPLES

EXAMPLE 1: $$ \begin{align} F_1 &= x^5 - y^7 \text{ (orange)}\\ G_1 &= x^3 - y^4 \text{ (green)}\\ \end{align} $$
EXAMPLE 1: $$ \begin{align} F_2 &= x^5 - y^{11} \text{ (orange)}\\ G_2 &= x^3 - y^4 \text{ (green)}\\ \end{align} $$
EXAMPLE 2: $$ \begin{align} F_1 &= x^2 - y^{5} \text{ (red)}\\ G_1 &= x^4 - y^7 \text{ (blue)}\\ \end{align} $$
EXAMPLE 2: $$ \begin{align} F_2 &= x^2 - y^{5} \text{ (red)}\\ G_2 &= x^4 - y^7 + xy^5 \text{ (blue)}\\ \end{align} $$

june 2021 - i think I'll try to do this again (PDF)

My universal pdf notes are getting confusing.

This post contains most of my pdf notes I have so far, so it's super long. Later posts are planned to be shorter (I hope).

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