# Julia Robinson and Hilbert’s Tenth Problem

At the turn of the 20th century, David Hilbert, a mathematician of German descent, published a most influential list of 23 problems which would capture the minds of many mathematical minds throughout the next 100 years and beyond. The tenth of these problems asked:

“Does there exist an algorithm to determine whether a given Diophantine equation has a solution in rational numbers?”

Firstly, you may rightly ask what a Diophantine equation is. To the best of my knowledge, this is just a polynomial (here in x, although more variables are allowed) like $p(x_1,x_2,\ldots,x_n)=0,\,$ with integer coefficients. In 1970,  a young Russian mathematician Yuri Matiyasevich concluded Hilbert’s question to have a negative outcome:

“No, there does not exist an algorithm which, given an arbitrary Diophantine equation, would tell whether the equation has a solution or not.”

But whilst Matiyasevich found the ultimate solution to the decades-long study of this problem, it had involved using previously pub;ished results by the brightest of minds – one of them being Julia Bowman Robinson: one of the first females to achieve notable recognition in the world of mathematics. And since we share the same day of birth (mine 75 years down the line), it seems fitting to write a couple of words about this inspirational woman.

The trailer for George Csiscery’s documentary on Julia Robinson’s life innocently appeared in my ‘recommended for you’ stash of YouTube videos- and boy am I glad that it did. Julia’s forte had always been in the natural numbers, and it was Number Theory which captivated her mind the most. She also dabbled in Game Theory and Sequential Analysis, when it seemed like her work on H10 was getting her nowhere. But all she had to do was wait for Yuri Matiyasevich to grow up!

Her own childhood had been tainted by scarlet fever, rheumatic fever (which disallowed her from having any children, sending her into a period of depression); Julia’s eventual death caused by leukemia. She thus suffered ill health throughout the duration of her life. But this didn’t stop her from marrying her professor, Raphael Robinson! I wouldn’t even dare to imagine marrying any one of my professors… *shudder*

“All this attention has been gratifying but also embarrassing. What I really am is a mathematician. Rather than being remembered as the first woman this or that, I would prefer to be remembered as a mathematician should, simply for the theorems I have proved and the problems I have solved.”

Indeed, Mrs Robinson was the first female mathematician to have been elected to the National Academy of Sciences in 1975, became president of the Association of Presidents of Scientific Societies (soon stepping down due to poor health). But her achievements as “first female” are not what she wanted to be remembered for.

Nonetheless, she is a role model. Having heard of Julia’s achievements of overcoming health obstacles and gender barriers, I feel like her and I would get along pretty darn well.

If only I were born three quarters of a century earlier…