Today in the history of science and technology | China's "mathematical twins" achieved important results in the study of function theory

47 years ago today, on February 28, 1977, Yang Le, an academician of the Chinese Academy of Sciences, and Zhang Guanghou, a Chinese mathematician, made important achievements in the study of function theory. They found the organic connection between the two main concepts in the study of function value distribution theory, "deficient value" and "singular direction" for the first time in the world; at the same time, their research on these two concepts also achieved important results at the world's advanced level. Their research promoted the development of function theory and was highly praised by the mathematical community at home and abroad.

◆“Mathematical Twin Stars” Promote the Development of Function Theory in the World

In 1925, Finnish mathematician Nevanlina created a theory when studying the distribution theory of function values. He believed that among a large number of common and important functions, the number of times the functions take each value is similar, with only a small number of exceptions. The values ​​of these exceptions are called "deficient values". The study of "deficient values" has become a major topic in the study of function value distribution theory. In addition, when studying the changes of a large number of common and important functions, in some parts of the range of independent variables, the functions take a lot of values ​​and the changes are extremely drastic. Mathematically, this phenomenon is described as "singular directions". The study of this "singular direction" constitutes another important topic in the distribution theory of function values.

Yang Le and Zhang Guanghou, a pair of companion stars in the mathematical sky, inherited and carried forward the fine traditions of the older generation of Chinese mathematicians. They integrated the classic achievements of the traditional French school of functional analysis, absorbed and learned from the advantages of functional analysis research in the United Kingdom, the United States and other countries, pioneered and proposed and solved some new international problems, thus forming their own style and research characteristics.

Yang Le and Zhang Guanghou took a unique approach and devoted themselves to research, and finally found that the two difficult problems of "deficit value" and "singular direction" are not contradictory or exclusive, but are mutually dependent, organically connected and mutually based unified concepts. Decades of hard work and wisdom have finally built a colorful bridge between the two long-separated research fields in the mathematics community. Their research has caused widespread influence at home and abroad.

On April 13, 1978, at the International Analysis Conference held in Zurich, Switzerland, Yang Le methodically gave an academic report on the fruitful achievements of Chinese scientists in the field of function value distribution theory in front of many world-renowned mathematicians. His report was warmly welcomed, and the participants praised it as "amazing results" and "wonderful report". The founder of modern function value distribution theory, Nevanlina, congratulated enthusiastically: "Just now you said that you came to learn from European mathematicians. I think that now European mathematicians should learn from you." Soon, the literature on mathematical research published in the United States fully quoted the theorems of Yang Le and Zhang Guanghou, saying that they were "both novel and profound" and were "first-class work" in pure mathematics, just like the research on the Goldbach conjecture.

◆ An experienced horse knows the way

In the early 1960s, after the implementation of the eight-character policy of "adjustment, consolidation, enrichment and improvement", the country resumed the recruitment and training of graduate students. When Yang Le and Zhang Guanghou graduated in 1962, it was just in time for universities and research institutions to recruit graduate students for the first time in the 1960s. After a rigorous examination, Yang Le and Zhang Guanghou became graduate students of the Institute of Mathematics of the Chinese Academy of Sciences, with Professor Xiong Qinglai as their supervisor.

In September 1962, Yang Le and Zhang Guanghou started their postgraduate studies at the Institute of Mathematics. At that time, the Chinese Academy of Sciences had just held a ceremony to celebrate Professor Xiong Qinglai’s 70th birthday. Xiong Qinglai said to Yang Le and Zhang Guanghou: I am over 70 years old, so it is difficult for me to be of specific help to you, but I am an old horse who knows the way.

In fact, the theory of function value distribution has a very long history. Many great mathematicians in the 19th and 20th centuries have devoted themselves to its research work, and there are countless outstanding achievements, important papers, and monographs. Xiong Qinglai first asked Yang Le and Zhang Guanghou to study "Picard-Borel Theorem and Its Generalization" by Nevanlina, the founder of the modern theory of function value distribution. Although the book is only more than 100 pages long, it gives a concise outline, allowing them to master the core content of the modern theory of function value distribution well and quickly move to the forefront of research. Then, Xiong Qinglai asked them to carefully study "Borel Direction of Meromorphic Functions" by the famous French mathematician Varianon, which is a very in-depth and difficult research field in the theory of function value distribution pioneered by Varianon. Xiong Qinglai pointed out a good direction and path for their research.

During their postgraduate study, Yang Le and Zhang Guanghou made remarkable achievements in the theory of function module distribution, angular distribution and normal family theory. For example, in the normal family, a research result obtained by Yang Le and Zhang Guanghou in the second half of 1964 has answered a research question raised by Professor Heyman, an international authority on function theory, at the same time. According to later information, Heyman collected and raised some important research questions at the International Conference on Function Theory held in London in September 1964. At that time, my country was in a very closed state. Yang Le and Zhang Guanghou did not know about the conference held in London, let alone the content of the reports and questions raised at the conference. However, their work at that time was able to satisfactorily solve an important problem of international concern, and the papers they published were constantly cited by some famous function theory experts at that time and in the following decades.

◆The "Zhang-Yang Theorem" was born

In 1964, Zhang Guanghou and Yang Le began to collaborate on the study of families of holomorphic and meromorphic functions. They developed a method of eliminating primitive values ​​and obtained very good results.

Zhang Guanghou and Yang Le collaborated to discover for the first time the specific connection between the two main concepts in function value distribution theory, "deficit" and "singular direction". For a long time, mathematicians have always believed that deficient value and singular direction are two completely different concepts in the study of function value distribution theory, and there is no connection between them. Yang Le and Zhang Guanghou's collaborative research revealed for the first time that there is a clear and close connection between these two basic concepts, and gave a quantitative expression of this connection.

This theorem shows that for most meromorphic functions (finite positive order), the number of their deficient values ​​can never exceed the number of their singular directions. For entire functions, the conclusion can be strengthened, that is, the number of their deficient values ​​can never exceed half of the number of their singular directions. They also gave examples to illustrate that the above result is the best. The result of Yang Le and Zhang Guanghou is groundbreaking and provides a new direction for the study of function value distribution. Their results are named "Zhang Yang Theorem" by the international mathematical community.

Then, Zhang Guanghou began to study the concepts of "deficient value", "asymptotic value" and "Julliard direction", which are three important concepts in function theory. After research, Zhang Guanghou successfully found the organic connection between the three, made a specific mathematical argument for this connection, and indicated the objective law reflected by the entire function or sub-morphic function. This is an important achievement of world level. In March 1973, "Science China" published a special supplement for the paper, and Xinhua News Agency and "People's Daily" also reported it again on the front page with the title "Zhang Guanghou has achieved world-class results again".