Chapter 451

Jacobian conjecture, English name Jacobian conjecture.

The simplest form of introducing the Jacobian conjecture is the Jacobian conjecture for two variables: two polynomials f(x,y), g(x,y) on a given domain k with a feature of 0, provided that the following Jacobian conditions are satisfied:

j_{x,y}(f(x,y),g(x,y)=non-zero constant in k.

Then we get k[x,y]=k[f,g].

That's what Jacoby's conjecture is all about.

As one of the well-known conjectures in the field of algebraic geometry, the Jacobian conjecture is not very difficult to understand.

Anyone with a little knowledge of calculus will be able to understand it easily.

Therefore, the Jacobi conjecture has always been in a hot field of research.

Since it was proposed in 1939, nearly a hundred years of development have brought mathematicians closer and closer to the true success of the proof.

Maybe four or five steps, maybe just one step.

Theories about the Jacobian conjecture have also been gradually refined, and some have even formed a separate research direction.

It can be said that in such an environment, it is only a matter of time before the Jacobi conjecture proves successful.

Last, someone pushed the door open!

And Cheng Nuo really wants to be the one who pushes open that door.

In terms of difficulty, of all mathematical conjectures, the Jacobian conjecture is only included in the third echelon.

The difficulty of similar ABC conjectures, Kakutani conjectures, and four-color conjectures in the second echelon is too high, and Cheng Nuo has not considered it for the time being.

After choosing, Cheng Nuo felt that Jacobi's conjecture was the most suitable.

Besides, since he had figured out a vague direction in his mind, Cheng Nuo thought he could try it.

If you want someone else to come, it is rare to have the opportunity to lead a topic, and you must choose a general difficulty to seek stability and accumulate qualifications.

But Cheng Nuo is different.

At most, he spent only half a year at Cambridge University.

It was useless to stay at the Institute of Mathematics and Physics, because he would not live in Cambridge forever.

Instead of a bland first half of the year, it's better to make a big one.

Jacobi conjectures that although it is not comparable to the few large mountains that have stood in the mathematical world for hundreds of years, it can barely be counted as a small mountain.

It just so happened that Cheng Nuo wanted to take this opportunity to test whether he could climb this mountain with all his might.

What a source of pride to engrave your name on the highest part of the mountain!

............

March 15th.

Within three days, Cheng Nuo's project application was approved and hung under the name of the Institute of Mathematics and Physics of Cambridge University.

Alding assigned Cheng Nuo an empty office on the third floor of the Institute of Mathematics and Physics.

The efficiency of the institute is extremely fast.

The next day, the funds, equipment, and personnel all arrived.

Cheng Nuo also met Denton and Joya in the office.

"Hello two seniors, this time it's not three days, but the next half a year, we will all have to fight side by side!" Cheng Nuo greeted with a smile.

Danton smiled bitterly, "Cheng Nuo, you still don't call us seniors." If you're bent on graduating, we're probably going to call you senior now. ”

Cheng Nuo shrugged his shoulders and didn't bother with this anymore, "I'll give you two a brief description of our research topic this time." ”

"The title of our research project is called "The Keller Mapping Local Bijection Method of the Jacobi Conjecture", which is simply ......"

After listening to Cheng Nuo's narration, Denton was speechless with horror.

His eyes widened and he swallowed hard, "Cheng Nuo, you have to prove Jacobi's conjecture!!"

Cheng Nuo grinned, revealing a row of white teeth, "To be precise, it's us!

"This is crazy!" Joya exclaimed, shaking her head.

They were only three students, but it was unbelievable that they were challenging the famous Jacobian conjecture in mathematics!

Cheng Nuo put his arm around the shoulders of the two of them, looked through the window, and looked into the distance, "Guess if you put it there, isn't it just for people to conquer it! And we want that person to conquer, why can't we?"

"We have a saying in China, 'If you try, you may fail, but if you don't try, you will never succeed'! Believe me, we will definitely be able to!"

"Hmm!" the two nodded heavily, their eyes becoming resolute.

............

Half a year later!

Spring goes and autumn comes.

On the path leading to the Institute of Mathematics and Physics, the breeze blows, and the maple leaves flutter in the air.

In an office on the third floor.

Cheng Nuo turned his pen, leaned on the back of his chair and looked out the window, looking at the autumn Cambridge University, his eyes were slightly distracted.

"Almost, it's time to get out of here!" Cheng Nuo sighed lightly.

Knock knock~~

"Please come in!"

The office door was pushed open.

Denton, who was wearing a peaked cap, held a stack of documents and put them on Cheng Nuo's desk, his face was full of joy, "Cheng Nuo, I have already shown our proof process to Director Aldin." ”

"Oh?" Cheng Nuo turned the chair around, touched his chin and said, "What did the director say?"

"The director took this document and looked at it for four days, and when he returned it to me, he didn't say anything, but ...... It's strange to look at me!" Denton said, scratching his head.

Cheng Nuo smiled, "The director didn't say that there was a problem with our proof process, right?"

Denton scratched his head, "Not really. The director said that our paper was perfect, and he couldn't find any flaws, so let's just submit it. ”

Denton continued, his expression full of excitement, "To be honest, I really didn't expect that we would really be able to prove the Jacobi conjecture in half a year!"

Cheng Nuo picked up the teacup and took a shallow sip.

Compared with Denton's excitement, Cheng Nuo seemed much more indifferent.

In fact, as early as about two months after the start of the research, when the derivation results came out, Cheng Nuo knew that his proof was in the right direction.

Now that I got this answer, I was in my expectation.

Since Alding said that there was no problem, there was no big surprise. After all, Alding has also been a review editor for the Journal of the British Mathematical Society, a mathematical journal in the country.

Jacobi conjecture...... I really proved it myself!

The expected excitement and excitement did not appear at this moment.

The process of coming naturally made Cheng Nuo seem to feel a little inflated.

Jacobi conjecture?

It's just a small level 3 conjecture.

The proof is out, what is there to show off?

It will be too late to be happy when the Riemann conjecture is proved.

"Is the pit occupied?" Cheng Nuo asked George with a smile.

George nodded, "I've already sent a preprint of the paper to arXiv." ”

"That's good. Cheng Nuo knocked his forehead, "But this paper should be submitted as soon as possible, the research on the Jacobian conjecture in various universities has never stopped, and it would be too uncomfortable if it was preempted by others." ”

"But what if you go to that journal?" Cheng Nuo frowned.

"Just vote for the Journal of the Yingguo Mathematical Society!" an old voice came. Director Alding walked in without knocking on the door, "After all, I am one of the review editors of this journal, and I write a letter of recommendation, which is estimated to get your paper published as soon as possible." ”

"I didn't expect it, I really didn't expect much, but I didn't expect you to really drum it up. Alding looked at Cheng Nuo with a complicated look.

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