"Oh!" Qiao Xi nodded lightly, her expression as usual relaxed.
Seeing her mother’s humble attitude, Qiao Yu continued to speak earnestly: "One of the core problems of algebraic geotry is to study the geotrical properties of algebraic varieties.
I only learned while working on this topic that previous research was mostly conducted over real or complex number fields, but if we switch to the p-adic number field, traditional tools can’t be used. It’s because the geotric properties in the p-adic field are more peculiar.
Let give you an analogy, tools in traditional complex geotry rely heavily on continuity and smooth structures, but these structures don’t hold in p-adic spaces. Do you understand? That’s where the value of Schulz’s research lies.
Okay, enough about this, just tell what you think after reading that paper?"
Qiao Yu generously waved his hand, sparing his mother from further criticism since she humbly acknowledged her mistakes.
"Hmm, after reading your derivation, I found it very interesting. If your proof is correct..."
"Wait... I need to correct you. You can skip that part; of course, there’s no problem with my proof! It has already been published in a top journal and validated by supercomputers."
Qiao Yu was displeased and interrupted Qiao Xi again; no matter what, even if it’s his mother, he couldn’t tolerate non-professional remarks about mathematics.
"Alright, alright, no problem with your proof. The geotric properties of curves seem to have a direct impact on the distribution of rational points.
If we combine it with the space you constructed, there might be a potential relationship between the geotry of algebraic curves and the distribution of rational points. Wait a second, I’ll get a pen and paper."
Qiao Xi then stood up; there was a ballpoint pen and a stack of paper printed with Yanbei University on the desk in the room.
Qiao Yu beca serious too, standing up from the sofa and moving to Qiao Xi’s side.
"Your previous conclusion is N(X)≤C(θ)=θ^g, which ans for any algebraic curve C, the number of rational points N(C) is influenced by both the deficiency and geotric constraints of the curve.
So if f(θ,g) is a function related to the geotric properties of the curve, could this function f(θ,g) approach a limit within algebraic curves that satisfy this geotric condition?
"In other words, there exists an upper bound where, with increasing deficiency, the number of rational solutions gradually stabilizes. So, I think N(C)≤f(θ,g)."
Qiao Yu stroked his chin, finding it very interesting.
If this could be proven, it would an proving an inevitable relationship between the natural upper bound of algebraic curve solutions and their geotrical properties.
Because it implies that as the deficiency g increases, the number of solutions might stabilize at a certain limit.
In layman’s terms, there’s a threshold; once reached, no matter how much the deficiency increases, the rational points won’t change because they are directly constrained by geotry.
In other words, Qiao Xi has proposed a very interesting mathematical conjecture.
If it could be proven, Qiao Yu thought it could provide a new mathematical perspective at the intersection of algebraic curve theory, number theory, and geotry.
Wait...
What new perspective, or not, did Qiao Xi really understand his paper?!
What kind of genius mom is this?!
"Hey... Mom, did you really co up with this yourself?"
"Well, after all, it’s the first paper you sent ; I tend to look through it when I’m bored. Then one day, I suddenly thought there might be this possibility.
Of course, I don’t know if it’s correct, and I don’t know how to verify it. But I thought you might find it interesting. If you have ti, you might want to find a way to validate it."
Qiao Xi pointed to the inequality she had casually written down.
"I don’t know, this needs proof. But the idea is quite interesting. Aren’t you busy with math exercises every day? When did you start researching algebraic geotry?"
Qiao Yu still felt sowhat incredulous.
Even if this is just a conjecture, without being able to understand his paper, it can’t be proposed at all.
For example, his Senior Brother Chen, even if he read his paper a hundred tis, probably wouldn’t co up with such an insight.
"You’re working so hard every day, I just thought it would be good if I could help you in the future, so I’ve been working hard recently. Besides doing so physics exercises, I’ve been looking at those mathematics books.
Although it’s difficult, it’s also quite interesting. Is it hard to prove?" Qiao Xi replied casually, then asked.
"This involves the relationship between geotric constraints and algebraic curves, which is more complicated. But you can construct a model first and do nurical experints and computational verification. If it aligns with the results, then it will be aningful."
Qiao Yu first professionally analyzed this conjecture, then criticized: "But aren’t you being too ambitious! Have you finished learning high school mathematics? And you started reading Schulz and researching algebraic geotry?"
Qiao Xi shook her head and said: "I’ve already gone through the high school math textbooks. And didn’t you send this paper? Weren’t you expecting to read it?
Moreover, to summarize algebraic geotry, it’s just describing geotric objects using algebraic equations. Curves, surfaces, higher-dinsional algebraic varieties, distribution of rational points, structure of singular points, and their relationship with moduli spaces.
As for Schulz’s paper, I don’t need to fully understand it. Just a rough understanding is enough. Besides, the perfect space he proposed is essentially about ensuring good geotric structures.
This way, the research objects in the space can be described completely. Actually, everything is interrelated, and the ultimate goal is to facilitate computation and classification."
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