Revisions to Standard conjecture on u-invariants?

Link to @FZaldivar's comment; link to article, while this is on the front page

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edited May 8, 2022 at 16:39

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This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)$-variable equations may not. Here is the Wikipedia explanation. In particular, although it seemed possible that the $u$-invariant of any field was a power of $2$, it was shown by Merkurev in 1991 that there is a field with $u$-invariant of any even number. But the hypothesis that it could never be an odd number was contradicted by Izhboldin (in Fields of $u$-invariant $9$) in 2001 who constructed a field with $u$-invariant of $9$.¹

Q. My question is: Where does this issue stand now, ~20 years later? Is there some sense among experts that there is a field with $u$-invariant for any odd number larger than $7$? Or is it completely open, with no prevailing hypothesis?

As FZaldivar pointed out in the commentscomments, it was earlier known that $u \neq 3,5,7$, so $u=9$ was the first realized odd number.

¹ "Oleg Izhboldin died tragically ... at the age of 37 after submitting this article" ZBL review.

This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)$-variable equations may not. Here is the Wikipedia explanation. In particular, although it seemed possible that the $u$-invariant of any field was a power of $2$, it was shown by Merkurev in 1991 that there is a field with $u$-invariant of any even number. But the hypothesis that it could never be an odd number was contradicted by Izhboldin in 2001 who constructed a field with $u$-invariant of $9$.¹

Q. My question is: Where does this issue stand now, ~20 years later? Is there some sense among experts that there is a field with $u$-invariant for any odd number larger than $7$? Or is it completely open, with no prevailing hypothesis?

As FZaldivar pointed out in the comments, it was earlier known that $u \neq 3,5,7$, so $u=9$ was the first realized odd number.

¹ "Oleg Izhboldin died tragically ... at the age of 37 after submitting this article" ZBL review.

This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)$-variable equations may not. Here is the Wikipedia explanation. In particular, although it seemed possible that the $u$-invariant of any field was a power of $2$, it was shown by Merkurev in 1991 that there is a field with $u$-invariant of any even number. But the hypothesis that it could never be an odd number was contradicted by Izhboldin (in Fields of $u$-invariant $9$) in 2001 who constructed a field with $u$-invariant of $9$.¹

Q. My question is: Where does this issue stand now, ~20 years later? Is there some sense among experts that there is a field with $u$-invariant for any odd number larger than $7$? Or is it completely open, with no prevailing hypothesis?

As FZaldivar pointed out in the comments, it was earlier known that $u \neq 3,5,7$, so $u=9$ was the first realized odd number.

¹ "Oleg Izhboldin died tragically ... at the age of 37 after submitting this article" ZBL review.

Changed link type to zbMATH Open zeview in order to possibly get track back

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edited May 8, 2022 at 14:32

Daniele Tampieri

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This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)$-variable equations may not. Here is the Wikipedia explanation. In particular, although it seemed possible that the $u$-invariant of any field was a power of $2$, it was shown by Merkurev in 1991 that there is a field with $u$-invariant of any even number. But the hypothesis that it could never be an odd number was contradicted by Izhboldin in 2001 who constructed a field with $u$-invariant of $9$.¹

Q. My question is: Where does this issue stand now, ~20 years later? Is there some sense among experts that there is a field with $u$-invariant for any odd number larger than $7$? Or is it completely open, with no prevailing hypothesis?

As FZaldivar pointed out in the comments, it was earlier known that $u \neq 3,5,7$, so $u=9$ was the first realized odd number.

¹ "Oleg Izhboldin died tragically ... at the age of 37 after submitting this article" ZBL ZBL review.

This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)$-variable equations may not. Here is the Wikipedia explanation. In particular, although it seemed possible that the $u$-invariant of any field was a power of $2$, it was shown by Merkurev in 1991 that there is a field with $u$-invariant of any even number. But the hypothesis that it could never be an odd number was contradicted by Izhboldin in 2001 who constructed a field with $u$-invariant of $9$.¹

Q. My question is: Where does this issue stand now, ~20 years later? Is there some sense among experts that there is a field with $u$-invariant for any odd number larger than $7$? Or is it completely open, with no prevailing hypothesis?

As FZaldivar pointed out in the comments, it was earlier known that $u \neq 3,5,7$, so $u=9$ was the first realized odd number.

¹ "Oleg Izhboldin died tragically ... at the age of 37 after submitting this article" ZBL.

This is well beyond my expertise, but I just learned some of the history behind $u$-invariants of fields $F$, where ($u(F)+1$)-variable quadratic equations always have a non-trivial solution, but $u(F)$-variable equations may not. Here is the Wikipedia explanation. In particular, although it seemed possible that the $u$-invariant of any field was a power of $2$, it was shown by Merkurev in 1991 that there is a field with $u$-invariant of any even number. But the hypothesis that it could never be an odd number was contradicted by Izhboldin in 2001 who constructed a field with $u$-invariant of $9$.¹

Q. My question is: Where does this issue stand now, ~20 years later? Is there some sense among experts that there is a field with $u$-invariant for any odd number larger than $7$? Or is it completely open, with no prevailing hypothesis?

As FZaldivar pointed out in the comments, it was earlier known that $u \neq 3,5,7$, so $u=9$ was the first realized odd number.

¹ "Oleg Izhboldin died tragically ... at the age of 37 after submitting this article" ZBL review.